Access is tough at Idle Hour. It's a very exclusive, thoroughbred-money club. It won't be inexpensive either. It is a GREAT course though. I think it's easily the best in Kentucky, and would jump at the chance if you can find a way on. I just think it's unlikely.
There are several great options in Lexington though. Stay away from the Marriott course. It's 10 ok holes followed by 8 of the worst holes I've ever seen. It's also much more expensive than other area public courses that are considerably better. Lexington's an excellent city for golf value, even if it doesn't have a great course outside of Idle Hour.
Here's my recommendations, in order of preference:
1. Champion Trace - A very good Art Hills course just south of the city. Private, but accessible with a friendly phone call and I'd be more surprised if they weren't willing to work with you. It's an excellent tournament course and quite a bit better than any other Hills course I've played. I swear I don't entirely mean that as a backhanded compliment. Great value at about $70 for guest play, give or take.
2. Old Silo - A bit of a drive, but not too far. Easily the best public in the Lexington area, and very good value. Can be played for under $50.
3. Kearney Hill - Nothing too special, but a nice PB and Pete Dye collaborative effort. It's hosted a few USGA tournaments and is a good, balanced test. Generally in good shape. Solid course. Under $50.
4. University Club of Kentucky - A nice 36 hole facility. I played a Ryder Cup style event there last year. Having two courses onsite was a nice perk. Some really excellent holes, but some awful ones as well. The home club of the University of Kentucky. Under $50.
5. Picadome - A nice muni in the heart of Lexington. Classic design principles on a tight property. 18 really good holes, with some pretty interesting bones for an architecture fan. Not the championship test of some of the others on this list, but still reasonably challenging at around 6500 yards. Small greens, some fiercely pitched. Quirky in spots. Could be pretty special with some tree clearing and a restoration/renovation that will probably never happen. You can probably play for under $40.
There are several other good courses in the area. Cherry Blossom in Georgetown is always in excellent shape, though perhaps a touch overwatered. Pretty standard residential design though. Same goes for Houston Oaks in Paris. Both solid courses that non-architecture nerds would probably prefer over Picadome, which is really more of a personal favorite for me.
The public golf in Lexington is far superior to the options in Louisville or Cincinnati, in my experience. There are good private options in all three cities if you can get access.