Shuttles are available all day and all night. It's very easy to find them: just walk out the door by the baggage claim, and they'll be there waiting for you. Shuttles to the Strip hotels are $6/person, but the Stratosphere is a little farther so you might pay a little more.
When you want to go back to the airport, the hotel can call a shuttle for you. If you take a shuttle to the hotel, they'll give you a business card and ask that you call their service for your return trip. 3-4 different companies run shuttles, and they don't want to take a chance on their competition getting your busines on the trip home.
The shuttles are slower than taxis (because the driver might stop at three different hotels to let people out before he gets to your hotel), but they're very dependable.
For company names and a for-sure price, google McCarren Airport, then check under ground transportation.