spiral up cut is the bit you want. i believe mine will cut up to 1". don't know if they make a longer one. just look around.
as for a router. im rocking a craftsman plunge base and it does the job just fine.
Go to MLCS Router Bits and Woodworking Products they have good quality bits for reasonable prices. Search eBay and pawn shops, should be able to find good quality used routers. Bosch, DeWalt, Porter Cable and Triton are what woodworkers consider the best available, consistantly get good reviews.
The quality of the router is of less significance than the precision of your jig. Pretty much any router can cut the rings, just have to make multiple passes going down about 1/8" each time you make a revolution.
Spiral bits will work better as they remove the waste from the groove being cut. Straight bits will work just the same but you will have to remove the waste yourself as you cut. Good Luck!!
How many do u really need to cut in 1 inch?
The dewalt will do a good job
And like others said its all about your jig and technique.
Also u may want to consider a table because I'd u get the one ring made and needed copy's it really easy to do
I have my plunge base mounted to the circle jig and the fixed mounted to a table