It appears you do have some experience. You've hand-fed and you've used a gavage needle (something I've never had to do, thank goodness). Also, you have easy access to an avian vet, which is a huge bonus! What I wouldn't have given for that so many times this year!
I hadn't hand-fed for roughly twenty years, so I had to build a functional nursery from scratch. I'll share what I'm doing in hopes it can help you. If someone else has a better way, let us both know.
As for a feeding schedule, my mother and I simply fed the baby when her crop was empty or nearly empty and let it empty completely once every 24 hours. It's good to do this to prevent crop stasis.
At three weeks old, my baby's crop emptied every three hours or so. It should take a bit longer for a four-week-old's crop to empty. Ideally, the baby should be weighed in grams in the morning when his or her crop is completely empty. If a young baby fails to gain weight, it's usually a sign something is wrong. It isn't unusual for a baby to stall or even lose a gram or two one day and then shoot up the next day, but if weight loss is accompanied by any symptoms, take it seriously.
Also, babies lose weight prior to weaning. My BFA (hatched May 1) was so plump, she started slowing down around the 7th of June in prep for the weight loss. For some reason, this scares me every damn time. XD How many times did I go through it? I don't know. Even so, be mindful at this time that there isn't something else going on besides a baby preparing to fledge.
If you have a good brooder, that's half the battle with a very small chick. If you don't have a good brooder...well, let's just say, I wouldn't want to go through those rough couple of weeks again! My mother and I nearly killed ourselves monitoring our much-loved baby blue front. We were her thermostat, constantly monitoring her around the clock. I'm
still tired from doing this!
With a four-week-old in a reliable brooder, you can sleep through the night. I would Google it to be sure, but I think a baby that age can go eight hours at night without food provided they are fed properly throughout the day (thick enough formula while still providing hydration). Once a baby has a good coat of down feathers, you can safely drop the thermostat to 85 degrees F or in the case of my very hot-blooded baby, 82-83. Let the baby, not a book, be your thermostat. A baby that is too hot will pant. If that happens, get it out of that hot brooder immediately! A panting bird cannot dissipate heat as well as a panting dog. Overheating is a bigger immediate danger to the chick than being too cool, although you should treat either extreme as the number one priority. A baby that is too cold will have cold toes and cold wing tips. If you're not sure, you can place the tip of a naked baby's wing gently between your lips. If the wing tip is cold, the baby needs to be warmed up. If the wing tip is warm and the baby isn't panting, it is at the right temperature. Babies that are too warm are usually red (this can also be the case if they're dehydrated); babies that are too cold are often pale.
A short time after my baby's wings were covered in contour feathers, it was plenty warm enough at my place to move her into the 20 gallon long aquarium she lives in now. She'll be ready for a weaning cage when I'm satisfied she won't break a blood feather on the bars.
At close to eight weeks old, I'm feeding my baby either twice or three times a day depending upon how well she is eating on her own that particular day. You'd think a very hungry baby parrot would eat solid food, but that's not how it works. Hand-feeding is a nurturing experience for a young parrot. After a hand-feeding, a baby bird feels safe and is therefore more likely to try new things. Some weaned babies who feel traumatized by their move to a new home revert to begging and do not eat solid food. It's not just the food they crave; it's the love. Sadly, this happened to a college friend of mine and her baby cockatoo died. I wish I had said something, but I wasn't feeling well at the time and I assumed the breeder would explain the phenomenon or my friend would realize something was wrong. I guess I ended up a little angry with both of them. This should
never happen to a baby bird.
The man who bred my little BFA uses three different formulas in case there's a bad batch. That way, the other two make up for the deficiencies of the defective formula. A fellow breeder I knew back in the day lost a clutch of blue and gold macaws to vitamin D toxicity. There was simply too much of it in the formula.

If you haven't hand-fed before, I wouldn't necessarily choose the shotgun approach. Why? If you choose a single formula, the directions are right on the bag, telling you how much formula to add to purified or distilled water. I
did use the shotgun approach:
four different formulas at the same time. My baby has very lovely feathers.

She's petite, but it's her natural size. I know this because she was a fat little thing before she started trimming down in preparation for flight.
A few times I decided my baby needed hydration. When I hand-feed warm water (always a small amount), I take care to do it more slowly and carefully than formula because 1) the baby is expecting formula and may gasp in surprise and 2) water is less viscous than formula. Obviously, you don't want the bird inhaling water, nor do you want her to lean against a crop full of mostly water and aspirate herself. I made decisions to hydrate based on the chick's color and whether or not she'd been too hot that day. If you have a good brooder with good humidity and you follow the instructions on a package of hand-feeding formula, you shouldn't have to hydrate the chick. Still, it's good to know the signs of dehydration. Sometimes I'd simply feed a thinner formula the next time. It was all a matter of past experience and observation.
Back when I was breeding, I had a device that would heat syringes to the perfect temperature. Now that fish tank heaters have auto shut off, I can't use that method anymore. Bummer. For this chick, I use a Ball jar with a handle for mixing the formula. Then I put the Ball jar with the mixed formula into a 1 Quart Pyrex container full of hot water. You'll get the hang of how much hot water to add. Just be careful for yourself and your chick when it comes to hot water! Now that my baby is older and more mobile, I don't get her out of her aquarium until the food and water are just the right temperature. I have two different temperature probes: one for the water and one for the formula. When they meet in the middle and go down as one, I feed when 1) the formula has reached the correct temperature according to the thermometers and 2) I've stuck my finger into the water to make sure it isn't too hot. While I'm waiting, I keep stirring so the formula becomes smooth and uniform in temperature. The breeder of my baby recommended I feed at 105 degrees, which works for most babies. Unfortunately, some babies want it even warmer than that and refuse to eat if it's not. (I'm looking at you, African greys!) Unfortunately, my baby wanted warmer formula. Joy. The breeder said not to feed over 107 degrees Fahrenheit, so I followed those instructions. Now that my baby is at the correct weight for fledging, her appetite is keener, so she might accept the cooler (and safer) 105 degrees again.
O-ring catheter tip syringes are the best syringes to use when it comes to feeding animals. They're even better if, just prior to use, you squirt your disinfected and clean finger with Pam olive oil and apply it to the rubber on the plunger of the syringe. Put the syringe together, pump it up and down a few times and you have a well-oiled machine. One of the main causes of aspiration in chicks is a stuck plunger that unsticks too quickly, dumping too much food into the baby's mouth. A bit of Pam olive oil will keep the plunger from sticking, especially when it comes to an o-ring syringe. Afterward, I clean it with dishwashing liquid and rinse thoroughly. I never had very many babies at one time, so I have always been a careful feeder rather than a fast feeder. As long as the baby is giving you a feeding response, add food. When the baby stops, stop. It may take awhile for you to get used to each other's way of doing things, but you will. I think it's best for the chick to actually taste the food. A "power feeder" can empty the entire contents of a syringe into a baby's crop in under a second. That's a handy skill if you're hand-feeding dozens or even hundreds of babies, but I'm not sure I'm brave enough to ever try it. It must be done when the baby is in the middle of swallowing or you will aspirate him for sure. So, as far as I'm concerned? No thank you. You have someone who will show you how, so I won't cover that.
I used to use Nolvasan to disinfect syringes because I often had babies from multiple clutches and I didn't want any cross-contamination. With just one baby, I use a baby bottle steamer that goes into the microwave. It works really well and means I'm not bombarding her little system with disinfectants.
For surfaces, I use white vinegar or Nolvasan S (2 tablespoons per gallon of clean water). Be careful not to spray where baby can inhale the fumes from the vinegar. The fumes are pretty rough! For that matter, I wouldn't spray Nolvasan near the baby either. Keep in mind that Nolvasan doesn't kill pseudomonas. I don't know whether vinegar does, but my guess is it does not. Pseudomonas is a very tough little bug that likes warm moist places, including the back of your parrot's throat. Keep surfaces clean to keep pseudomonas at bay.
You said you already got your supplies. If you like any of my ideas and would like to know where I got my supplies, just ask. I hope I was of some help.