Forget about the first weekend in May - Alyeska is closed that weekend.
This is the last week of 7-day skiing, then they go to weekends (Saturday and Sunday ONLY) through Memorial Day weekend.
But, May 5 & 6 - closed. They always close the first weekend in May, at least that's what patrol told me last year, and is what they are reporting on the web site & the daily ski reports as of now.
The rest of the weekends in May, open Sat & Sun 11-5. Chair 6 only. No outer areas. High Traverse is already closed for the season, and this weekend is the last weekend for North Face. No lower mountain either, just the top 1400'.
The ski school & ski rentals at the ski area are typically closed as well by the end of April/beginning of May. I'm not sure where you could get rentals (maybe World Cup is still open? Or someplace in Anchorage?) if you needed equipment.
The spring skiing is fun, I won't deny it, but I wouldn't make a trip up here just for that, given the limited terrain and hours open in May. If you happen to be coming up to visit AK & want to hit the ski area, great, but I wouldn't recommend a special trip just for skiing the resort this time of year.