MTB 9 speed sram shifters/rear deralier, shimano 9 speed cassette (HG?) only shifts to 8 cogs
If you've already checked the lower limit screw of the derailleur, and also checked for a bent derailleur hanger, another thing to consider is cable friction. To get to the smallest cog, the derailleur spring has to move the derailleur all the way to the outside. While it's doing that, the spring is pulling the shifter cable out of its housing.  If there is a lot of dirt/oil/gunk buildup inside the cable housing, that would make it difficult.  

Given that the bike is fairly old, it's worth removing the shifter cable and lubricating the inside of the cable housing. Then you could either reuse the old cable if it's in good shape, or better yet, get a new shifter cable (should cost just a few dollars).