Honestly, I would just break the tank down and wait until you have the funds to do the tank the right way. It will come out so much nicer and healthier that way. That is what I am doing with my 210. I have the money to afford the tank, stand, and sump, but I know I will not have the rest of the estimated $3000 until the end of July so I am waiting until then to start it up because it will be so much better than if I were to start it up now with hardly anything.
However, I know most people would not want to hear that and that was not the original question, so to answer the original question, no, do not run the skimmer during your cycle. I would just let the tank cycle out, and once you have no ammonia and nitrite, turn the skimmer on to help bring down some of the nitrates.
If you will take any advice though, introduce livestock VERY SLOWLY. If you can leave the damsels in there a few months after the cycle, that would be best. Just let them establish some bacteria while your bioload is not very high, and once you have the money for 30-45 lbs. of live rock, cure it and introduce that before getting the damsels out and introducing new fish. Even though it is fish only, ALL systems benefit greatly from live rock and live sand.
Good luck!