If you've never had one before, it might be better to start with a red dot, particularly if money is tight. I'll recommend the Bushnell TRS-25. It's a good red dot sight at a reasonable price, I have one on my S&W M&P 15/22 and I like it a lot.
If you're going with a scope - as others here point out correctly, you're going to have to do some homework if you want it to fit in the bag while still attached to the receiver, and you have to think carefully about positioning and eye relief. If you don't mind detaching the scope and you have a quick-detach setup, then you can use almost anything that gives you the right eye relief. If you want it to stay on the receiver and go back into the bag, your choices are more limited. The whole thing really boils down to whether you think it's more important to keep the scope attached to the receiver and still be able to fit the buttstock portion back into the backpack.
When Jeff Quinn did his review of the 10/22 Takedown back in 2012 he had a Leupold 1.5 to 5 power VX-III mounted. That's a pretty expensive scope but you might be able to find something closer to your price range with virtually the same dimensions...I just wouldn't expect it to be as good as a Leupold.
http://www.gunblast.com/Ruger-1022TD.htm
My feeling is that for a relatively short-range cartridge like the .22LR a decent 3MOA red dot will do a lot of good, and the TRS-25 is a very small and lightweight assembly.
