Depending on how much you care about keeping the uke in good shape, either a soft or hard case is a good bet. A gig bag is not going to protect against the abuse that being tied to the outside of your pack will cause (this will probably be the best way to carry it as it will eat up
lots of space on the inside). As an example, check these out (right click to open the link in a new tab/page)
upgraded-ukulele-cases
Nothing will be waterproof, but that's easily solved by placing the case in the bottom of a clean garbage bag, rolling it like a burrito, folding the open edge of the bag like an envelope and taping it down with duct or packing tape. You can also make "straps" with the tape to use to tie the bag to your pack. You might get several uses out of this waterproof method if you're careful not to tear the bag when you open it up. BTW, take your own tape and bags, availability on these could be sketchy. Don't let whatever case you buy get wet, they never, ever dry right and expose the instrument to way too much humidity.
Don't worry about getting a too large case, you'll need room for music, extra strings, a pick and strap if you use one and you'll have extra room for whatever doesn't fit well in your pack (use socks, etc. to pad the uke in the case). Depending you your airline, you might be able to take the uke/case as an uncounted carry on bag since it is a musical instrument. Ask first, though, and if you do have to check it, use the waterproofing method, lots of padding all around and labels everywhere with your itinerary and contact information (Joe Smith / DFW-MAD / AA 36 / 01-01-21 / email address / (111) 123-4567) plus what the luggage check attendant will put on. That way it'll at least end up at your final location or get re-routed back to your origin if it gets lost and you can pick it up at some later time..
Granted, not the easiest logistics to deal with and, honestly, it'll probably take some knocks, but it's better than carrying a double bass. I take an harmonica these days. Good luck, play and sing lots.
[Edited to remove email address, even though fictional]