Just had a look at the anthea apartments website but nothing about a gym.

http://www.antheahotelapartments.com.cy/article.php?id=66

Ive stayed at the anthea on 2 occasions and found them to be one of the best appartments in napa although i dont remember them having a gym , although both times were a few years ago ,there are gyms in ayia napa though , there is one not far from the center im sure if you ask any of the bar guys they will point you in the right direction