I understand Jack Bros. is a popular and rare cartridge, are the US and J versions equally rare and equally expensive?

EDIT: I found the answer on the Jack Bros. page, J: 4/10, U: 7/10.

What about price, what could I expect a Japanese version to end at on an auction?

I got mine for ¥ 1200 in Tōkyō. So pretty much you’re being ripped off if you pay much more for the game itself (without s&h).
I don’t know about the rarity, but it seemed pretty common for game shops that are into selling retro to have Jack Bros., Galactic Pinball etc. I even saw two of the rarest games (Space Invaders and the Super Robot one) in two different stores.
So yeah, eBay is pretty much for people who want to get ripped off nowadays it seems 🙁

cYa,

Tauwasser

But it’s probably cheaper buying off eBay than take a trip to Tokyo.