If you don't mind a long day, the tour to Tulum is quite fascinating. I definitely do concur with the recommendation to stick with the ship's tour due to the time and logistics involved. If the tour runs late, the ship will wait for it or, if it misses the ferry connection, make the necessary arrangements to get the passengers back aboard. If you go on your own and run late, you are on your own (and please excuse the pun).
If you go to Tulum, be sure to take a large (liter) bottle of water from the ship for each person in your party and be sure to drink from it frequently, especially while visiting the ruins, to prevent dehydration. The climate there is very warm, with normal daytime temperatures in the 90's even in the middle of winter.
If you prefer a less ambitious excursion, there are some Mayan ruins at San Gervasio on the island of Cozumel. The ruins there are less extensive than the ruins at Tulum, but they are more significant because the island of Cozumel is sacred in the Mayan religion.
There's also a relatively new archeo-ecological park at Ex Caret, also on the mainland, that supposedly has some excavated ruins in addition to a lagoon for swimming and snorkeling. I don't know too much about it, but I again recommend taking the excursion from the ship rather than going on your own due to the logistics involved.
Chitzen Itcha is a long way from Cozumel and Playa del Carmen. Some cruise lines do offer excursions there that do indeed go by air -- and they are very pricy.