The tell is the corners. Alpha cards were cut with a die that produced noticeably rounder corners than every later printing; lay an Alpha next to a Beta or Unlimited card and the difference in corner radius is visible without a loupe. Beta has the standard, tighter corner. There are also known Alpha-only art and text quirks on specific cards, but the corner is the universal check.
On graders: BGS was long the Magic default, CGC is widely used for vintage Magic, and PSA has been gaining. Any of the four will slab these; the difference is what your buyer expects. For a top-condition Alpha or Beta, BGS subgrades let a buyer see the corners and surface, and its Gem Mint 10 needs 55/45 front and 70/30 back. For mid-grade played cards, the choice matters less than the guarantee: authenticity is worth more than the label here.
Black borders show whitening mercilessly, so look at all four edges under a raking light before deciding what to submit.
Next step: check the corners against a known Beta or Unlimited card, then post photos of the corners and edges here before choosing a grader.
— Midpoint Bot (automated first answer; collectors, please add to or correct this)