We can make a few rough estimates of his age, but there isn't enough information/consistency to get an exact number...
First consider the Decapodian life cycle:
These stages were introduced in the episode "Teenage Mutant Leela's Hurdles" where the crew is "de-aged" due to the effects of chroniton-infused tar and oil-eating bacteria. Unfortunately, the rate of de-aging doesn't tell a lot about Zoidberg's age relative to the other characters since it is highly inconsistent across the characters: The Professor seems to get younger at a faster rate than the others, while Hermes (who is about 20 years older than the others) ends up being about the same age as the rest of the crew. Add to this the fact that chronitons can cause random jumps in time and can exhibit highly localized effects and there's no telling how much each character was affected.
The one thing we do learn with certainty from this episode is that the Cuttlefish Stage of Zoidberg's development occurs when he is a teenager:
Zoidberg: Hooray! I'm a teenage heartthrob again!
Further, we also know from the episode "A Taste of Freedom" that the Humanoid Stage of development starts with Zoidberg looking like a small child. If we assume that growth from child to adult during this stage follows the same rate as it does in a human (along with the amount of schooling) and that it takes about 4 years to finish college and 5 more years to get a doctorate in art history, then we can estimate that by the time Zoidberg and the Professor first meet Zoidberg must be at least around 40 years old already.
This would place Zoidberg's current age at around 125 (rough minimum). So why might Harold Zoid look so much older after 80-or-so years but Zoidberg doesn't? I can only speculate that the Humanoid Stage of Decapodian development is very long with a prolonged stage of "middle-age" where aging isn't apparent. Harold Zoid could have been near the end of his middle-age stage during the 2920's, with the effects of aging subsequently setting in over the next 80 years and making him look much older.