The NPS makes clear that the ABSOLUTE EARLIEST the Going to the Sun Road will be fully plowed will be 2013 June 21.
http://www.nps.gov/glac/planyourvisit/gttsrfaq.htm
You will be able to see SOMETHING along whatever part of this road has been cleared (and "how much" will be cleared is impossible to predict, even on May 1), but it won't be the best possible time to make a visit.
The NPS notes that the visitor centers will open in early May:
http://www.nps.gov/glac/planyourvisit/hours.htm

May is way too early. In 2011 the road did not open till mid July as it was a high snow year. As of today, we are experiencing a mild winter with low snow but that could change easily with a few heavy snowstorms. Mid june is a pretty good bet at this point in time.
In May, the road will be open from the west entrance as far as McDonald lodge ( as it is all winter ) and on the east side , normally to Jackson Glacier overlook. To get from the west entrance to the east or vice versa you will need to go around U.S. Highway 2 which is at the extreme southern end of the park. The Many Glacier area and the Two Medicine area most likely will also be closed.
If you can postpone your trip a bit I would sugesst doing so to take full advantage of the entire park.

Rocky Mountain National Park generally has its main roads plowed and opened Memorial Day. It is also a spectacular park.

Have you been to Santa Fe? Spectacular mountains and much else. Less pie than Montana but better New Mexican, Tiwa (American Indian), and Mexican food!

On June 1 2012, I drove up from St. Mary 17 miles to where the gate was closed. Logan Pass was opened about 2 weeks later.