Units of measurement in a non earth centric universe.

[farntfar]

I really can't see a problem with a story that speaks of Tuesday, or the 10th of August. It will be much clearer to the reader than getting all bogged down in some absolute timeframe that doesn't really exist.
Ans as for measurements, the problem is simply that they (particularly the meter and the second) must generally relate to a planet. If it's not going to be Earth then it must be Trantor.
Yes of course you can define it as however many wavelengths of the emission of a photon produced by the transition of an electron of the atom caesium whatever from level X to level Y. A remarkably impractical measurement on a daily basis if extremely accurate.
It's not the sort of thing I can use when measuring pieces of wood to put up some shelves. A fairly accurate tape measure will work much better for that whether I'm in France or on a gas giant in orbit round a star in Ursa Minor.
I also have a laser measure which is supposedly pretty accurate. And most travelling spacemen aren't going to need anything more accurate than that.

 

[Vertigo]

My years are cycles and days are spins (of the planet).

I'd be a little cautious about using a word like 'spins' that refers to the spin of the planet. It is almost certain that any civilisation's word for a day would have appeared and become standard use long long before they would have understood they were on a spinning planet. In fact in general that sort of consideration should be applied to any measurements that would have preceded the knowledge of where that measurement actually comes from. So, for example, with time it is likely a civilisation would first start with the length of their day and then started dividing it down as we did into hours, minutes and eventually seconds. It was only much later that we eventually tried to find a reliable natural measurement with which to firmly calibrate those seconds.