... i had a parakeet when I was a kid about 40 years ago. Back then we were told not to use a grid in the bottom. They had a paper that I believe had some sort of sand glued to it. i can tell that things have changed a lot since then.
As Noodles noted, avoid the sand. No sand paper (budgies do not need grit), and no sand perches. Regarding the grate, I leave mine in, and originally I put paper Under the grate, but now I put it on TOP of the grate. You can use clean white paper towels (although that can get expensive!), or any newspaper. (I take a handful of grocery flyers on Saturdays, since my store puts new ones on Sundays. DO sniff any printed paper you use, and make sure it's not giving any fumes or odors.) The Budgie most Likely will Not chew the liner paper. (IF it does - then you will want to use only white or only black-white printed.)
The budgies generally have less drive to shred their lining paper than some of the larger bird (such as Noodles' 'too). Putting the paper OVER the grate will make cleanup a Lot easier for you - just change the paper daily. Then you will not need to clean the grate & the drawer of the cage daily.
Also make sure you change the food daily, even if it Looks like it's full. They leave the husks behind which float to the top of the bowl, making it Appear full, and they can actually starve to death quite easily. (Well, you probably know this since you had budgies before.) Also make sure to add in the pellets SLowly. If the budgie is suspicious of the pellets, it may pretend-eat them... and again, can starve. However it is best to start that changeover now, as it is new environment, probably hungry, great to begin introducing the pellets now.
I think someone else has mentioned this already, but, IF you or family member do keep the birdie, you will want to get a checkup with an Avian Vet if possible.
Good luck with the budgie. If someone is looking for the little birdie, then I hope you find the owner. Otherwise, I'm glad it found you, and best of luck with getting it settled and happy and healthy! Thank you for caring for it.