What does Cycles mean?

An instruction cycle (sometimes called fetch-and-execute cycle, fetch-decode-execute cycle, or FDX) is the basic operation cycle of a computer. It is the process by which a computer retrieves a program instruction from its memory, determines what actions the instruction requires, and carries out those actions. This cycle is repeated continuously by the central processing unit (CPU), from bootup to when the computer is shut down.

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Exactly how long a cycle is depends on the computer and the data that is being processed.

I believe, given the reference in the question, he is referring to years. Because it was built 28-29 years ago and Flynn had been trapped in the computer for 20 of those years.

A cycle is 5.745 days