Farscape was intentionally ambiguous with the precise lengths of its fictional time entries, for much the same reason why Star Trek is ambiguous with what exactly a "stardate" means. However, there are two likely possibilities.
One Cycle could be 366 Solar Days.
The Farscape Solar Day is approximately equal to one earth day, or about 24 arns / ~64,800 microts.
Given that the lesser units are very close to human units (for reasons both meta and story), it isn't a stretch to assume that the cycle was similarly linked to the human time scale. The simplest measure would be to round the uneven number of days in a solar year up (since it's "slightly longer" and not "slightly shorter", we know they didn't round down.) The only question is how far they rounded.
A value of 366 solar days per cycle results in our sample levithan lifespan being 109,800 solar cycles, which is only 225 more than the 109,575 in 300 years.
One Cycle could be 360 Solar Days... with longer Arns.
While an Arn is "about" an hour, a Solar Day is "about" a day, and a Microt is about 4/3 of a second, we don't know how close the variance is for the other items. And, in truth, we don't even know for certain that a Microt is 30% longer than a second; Crichton could have easily adjusted his estimate, rather than do a strict conversion.
It's easy to postulate that the numbers are slightly off in either direction, with a preference for easy divisibility not dictated by adherence to either Terra's solar and lunar cycles or human history. 24 is a good number of segments to divide a day into, as you can evenly split it into 2, 3, or 4 equal periods, and stagger such with effective overlap if you need to. Not so 366 segments of a year, which can only be evenly divided by 2, 3, or 6.
A 360 "day" cycle gets you divisibility by 7 of the 9 single-digit counting numbers after 1, and 12 of the whole numbers between 2 and 20. (There are reasons beyond historical inertial why we use 360 segments to divide circles, after all.) And if a "year" is 360 solar days, and a solar day 24 arns, we are left with a need for a longer arn if we aren't going to lose about 2% of the duration per year.
If you take the 360 - segment idea and carry it downwards, with segments of 60, you can mimic earth-time with a value shift of about 1.014583 for each inferior unit. Increasing a second to a 'microt' that's 1.5% longer, with the same increase for "hours" of about 3,652 seconds (Up from 3600), you wind up with a "year" of about the same length with far easier divisibility.
The precise value-shift would depend on how accurately you define a year -- is it "365.25" earth-days, or "290,097,396,344,952,000 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom." (i.e., rounded up form the SI Second.)