I'm a pianist/singer on NCL, so it's very different than your situation (while I get many requests, if I don't know something I just say sorry and/or try to play them something related), but if you're leaving soon, here's what I'd do: make sure you have recordings of the 250 songs -- you can probably find them on youtube, and then you can make mp3s with a converter such as clipconverter.cc -- you will not be able to easily obtain music while onboard due to slow and expensive internet, so you'll want to have copies on your computer/device. Your bandmaster may have them and be able to give you a key drive or something, but best to be prepared on your own. Hopefully they'll have a book of charts for you as well, but it's always good to have definitive versions available for listening.
If the 250 songs includes lots of standards (likely), then you'd probably only be in trouble if your resume/application said something like "have excellent knowledge of standards", or if you submitted a song list that contains tunes that you don't quite know. If you're singing in front of the band, they probably won't want you to use lyrics, but I've never been on HAL so can't say for sure.
A typical day for the first couple of weeks will include lots of safety training, so it can be pretty intense if you're also rehearsing and performing. After the first couple of weeks you'll probably be free all day, which will allow ample time for music study. Here's to a great contract, and may your voice never tire!