That's a tough choice! A Hi Wall action is going to be hard to find at a reasonable price, so you've really restricted your chances of building a budget gun right off.
I search quite often for these, and have only come across a few over the years that were usable as is to build up a gun.
As for calibers and barrels. I'd suggest either buying a Green Mountain barrel, or looking for an original takeoff barrel. The .22 Hornet would be an excellent choice, if small bores are what you're interested in. The GM barrels go for just over $100, so they're reasonable, and delivery time is fast!
If you were to consider a Low Wall, it would not only be more reasonable, but actions are much more common to find. I see complete "sporterized" rifles in my area a few times a year, in the $250-$300 range. The few Hi Wall actions I've seen were all over $600, and they were only actions needing work.
Hope this helps.
