The oldest continuously operating casino in Las Vegas is the Las Vegas Club, which opened in 1938 in its present location under the same name. Of course, it's been through countless renovations and expansions ever since, so the original layout has been long lost.
The El Cortez opened downtown three years later, with 79 rooms. Unlike the Las Vegas Club, the EC has been expanded only once (in 1983) and the original brick structure (the lowrise casino wing on the corner of Fremont Street and Sixth) was retained intact. Other than a new carpet once every few years, this wing is in its original state even today, 63 years later, making it the oldest original casino building in Las Vegas.
The oldest continuously operating casino on the Strip is the New Frontier. Again, it's been through at least four major incarnations since it opened in 1942, including two complete demolitions and rebuildings and four names (Last Frontier, New Frontier, Frontier, and new New Frontier).
