We happen to have a Las Vegas phone book from July of 1989, and Miss Lucy's is not listed in that phone book...which doesn't mean that it didn't open later in the year.

I was able to find a photo, courtesy of Roadside pictures, that shows the building number as 129 (the arrow on the photo points very close to the number).

I was also able to find a clipping from Newspapers.com that shows that Miss Lucy's was open in 1990. (3rd column)

Sorry I can't give solid answers on the dates, but maybe someone else has more clues