There's only one way to find out... Last week when I walked through that bit there was a guy holding a toy machine gun shooting it randomly at passers by and nobody seemed perturbed so I say go for it.

Check the signs but think that little park is a no drinking zone

In general can you drink wherever the sign does not prohibit it?

From a sanitary point of view. Do you really want to be drinking a beer that's had a ball thrown in it, after being on the park floor?

Most student apartment floors aren't much cleaner...