I have taken that ferry (the Blue Star Line) in Mid- to Late May 11 times since 1999. The schedule has not changed by a minute for decades. I have always purchased the "economy" ticket from Santorini-Naxos the day before... it's just "first come first served" for the best deck seats. On the Naxos-Pireaus leg, I buy my ticket in Naxos, maybe 3-4 days before sailing. The only time that ferry is at all full in Mid-Late May would be the 2 Sunday Sailings back to Piraeus.
Blue Star's large modern ferries accommodate between 1200 - 1800 passengers, depending on the model. This is a so-called "conventional" ferry (altho it's considerably faster than "conventional ferries" in other fleets). It has large capacity, carries cars & trucks on bottom deck, and has ample outside decks for the view; thus it is obviously the best one to take to Naxos to see the famous Santorini caldera view on departure. Costs about €17 - 19, last I checked; takes about 2 hours.
The "fast-ferry" accommodates about 450 passengers, is enclosed, has no view except thru spray-coated windows, costs almost 2x as much, and only saves about 15 minutes on the Santorini-Naxos leg. For the Naxos-Pireaus journey, Blue Star takes 5 hrs 15 minutes; the "reserved seat ticket" is now about €36 ... the fast ferry may save an hour but costs about 2x as much.
One interesting thing to consider, if you want to squeeze as much "beach time" as possible into your holiday. The earlier daily sail from Naxos leaves 9:45 AM - arrives 3 pm, thus chopping out the whole middle of a day. The 2nd "sunset sail" leaves 6 PM - arrives about 11:45 @ Pireaus. Of course, you must leave your hotel room at 12 noon normally. However, you can usually arrange with your hotelier to stash your bags behind reception and then, in late afternoon, have the use of a vacant bathroom to wash up quickly, refresh yourself & get ready for travel. Thus, you will have a whole extra afternoon to swim, sun or wander around town.