I think you could do it for $1,000 if you do like you say and plan carefully. Two important points - that is assuming you have the gear you will need and don't have to buy a lot of stuff. Secondly, definately find someone from your part of the country to hook up with and share travel expenses. Not only will it be cheaper, but you will enjoy the trip more with a good freind or two to share your daily adventures with.
If you are in good enough shape to consider it, a backpacking trip could get you into some very good elk country. If you hunt from a camp you can drive to expect to see plenty of other hunters.
Start by looking at where you want to go. Spend a bunch of time looking at the States websites to get information on tag availability, herd densities, hunter success rates, and availability of public land. Contact as many people as you can at State game agencies, sporting goods stores, here on Trad Gang, and start putting the puzzle together. Many guys here will help with some information if asked on a private message but don't want to post to much information for public veiwing.
I say do it! I've done that type of trip before, and no, we weren't successful at bringing home the meat. But we sure had a wonderful time and I don't regret going in the least. A trip to the Rocky Mountains is a magical experience to those of us who don't live there. Just the trip itself will make it all worthwhile. And with proper planning it is possible to have a great hunt too! Remember, the planning and dreaming is all part of the trip and is every bit as enjoyable as the trip itself.
Don't wait to get old to do this trip! If it has always been a dream, make it happen!