When was this Lego Adventurers T-Shirt produced?
The shirt itself says 1998 on it, which coincides with the earliest known release of a set for that theme.  



Here it is advertised on/in this 1999 January/February issue of Lego Mania magazine:  


(https://www.flickr.com/photos/timlydy/22805063335/in/photostream/)  

Given this was advertised at the very beginning of 1999, I think it is safe to assume that 1998 is the correct date for when the T-shirt and Magazine were printed. As for a model number, as Alexander O'Mara mentioned, it's listed on Bricklink as TS10.  I am not familiar with model numbers for Lego clothing, or if they even have them.  Bricklink has likely created their own format for designating a number to a particular design to simplify and clarify it's identification for buyers and sellers.