I've never stayed at the Brown, but have hosted many out-of-town guests in a professional capacity there. As others said, it's actually pretty reasonable as long as you're not there during the Derby(good luck getting any room-even at a roach motel-for a reasonable rate then). IMO, it's worth the experience, especially if the tournament is at the International Convention Center or the Yum! Center. The Galt House is also a good choice and a little bit closer to either of these locations.
If the tournament is at Freedom Hall/the Fairgrounds, probably your best nearby bet is the Crowne Plaza(old Executive West). It's been renovated in the last few years, and is quite nice as well as being reasonable. It is-quite literally-next to the front gates to the fairgrounds-within walking distance if you're so inclined. It's also just a couple of miles from the Airport.
While in town, I'd suggest a trip out to Knob Creek if you have the time. Even during the non-machine-gun shoot times(which I avoid due to the traffic), it's still a nice and well stocked gun store as well as a decent range. If you go out there, stop at Biff's Gun World-it's an experience like no other. Biff is actually a part owner of Knob Creek, and the father of Kenny Sumner who runs the day-to-day operations at Knob Creek. I've been going to Biff's for a while, and I find something new every time I go. The store is stacked floor-to-ceiling with merchandise(not just guns, but everything else imaginable), and with aisles that make it difficult for two people to pass.
On the other end of town, Shooter's Supply is obnoxious and high pressure but has a great powder selection. Lotus Gun Works is a very nice, well presented store that occasionally turns up interesting items, but is often a bit pricey. 111 Guns on Hurstborne is a smallish store that usually has a decent selection and a few interesting older guns in the back room.
Avoid Tilford's, but River City Firearms just up the road is a good-albeit small store.
Cabela's is worth a visit-when I was in there yesterday they a lot of nice high-end doubles back in the gun library. Mike-the gun library manager-is a good guy and will take good care of you.