Yes, lime is to make soil alkaline. I think you should consider other
crops because the potatoes will be miserable. Changing the pH on the
fly is easier said than done and you may ruin the whole plot for a
while by applying too much sulfate. Onion, chard, beet, and just about
any brassica (broccoli, cabbage, collard, kale) will grow well in
alkaline conditions. Kale will give you something edible in six weeks
if the weather cooperates.
You still have to add a little organic matter and fertilizer to make
them happy. If you apply manure, it will provide both. In your
conditions, one of the best organic amendments are wood chips, that
will lower pH (they are acid), give you good mulch and good humus. You
have to add some nitrogen fertilizer (a single bag will be good for the
whole school garden).
On the other hand, wood chips are OK only if you plant little plants or
big seeds like squash. You cannot seed directly things like kale (many
small seeds) on wood chips until they have decomposed.
Make a Google search and you should find a few more crops that will be
happy above 7.0. I am guessing, for example, that lettuce, garlic,
beans, peas and tomatoes will manage with that soil. Various squashes
and cukes might be a little less happy but still manage. Potatoes are
really the only big no-no.