The oldest original building in Las Vegas (and the state) is the Old Mormon Fort, built in 1855 (and today a 2.8-acre state historic park located at 500 E. Washington Ave. about a mile north of downtown); see QoD 10/09/05 for the whole Mormon Fort story.
The oldest original building since the city of Las Vegas was established is the Golden Gate hotel-casino (then known as the Miller Hotel), at the corner of Fremont and Main (1 E. Fremont St.), dating back to 1905, exactly 50 years younger than the Fort. Though the building has been remodeled many times over the years, the exterior was restored to its original condition and the hotel rooms are still the original (small) size and layout. See QoD 2/15/08 for that history.
The oldest original casino building in Las Vegas is the El Cortez, which opened at the corner of E. Fremont and Sixth in November 1941, 36 years after the Golden Gate. Of course, the interior has been remodeled many times since, but what you see from the outside is exactly how it looked 68 years ago.
The oldest building on the Las Vegas Strip is, apparently, the Diamond Inn Motel at 4605 Las Vegas Blvd. S, built (we believe) as a private residence that was incorporated into the original Desert Isle Motel sometime in the early 1950s; see QoD 8/12/08 for that whole business.
But to finally answer your question, the oldest Strip hotel still extant is the Riviera. Actually, the Sahara predates the Riv by three years, having been built in 1952, and though some of the original casino might have been incorporated into the many expansions of the Sahara over the years (notably a 14-story tower in 1966, a 24-story tower in 1968, a 26-story tower and meeting area in 1988, and a 600-room tower in 1990), the original low-rise garden rooms were razed during a casino expansion (along with adding a new pool area, indoor racing-themed wing, and roller coaster) in the 1990s.
The Riviera opened in April 1955 with a nine-story high-rise (at the time, the tallest building on the Strip) and today that T-shaped wing, the Riv's closest building to the Strip, is identifiable from, for example, the Stratosphere Tower; you can see the original building below a few-story addition on top of it and surrounded by room expansions dating from 1967, 1975, and 1988.
The Tropicana's garden rooms occupy the second oldest buildings on the Strip and the only low-rise hotel rooms left from the garden-room days of the 1950s.
