Hiya Bilbow,
I do this grid every week and thought this clue was one os the stars this week.
You need to split it up at the right place and read it as:
First thing for Harvard - The setter is very shrewdly giving you the Def with added information which must be read slightly different because the 'Harvard' ref is there to let you know the answer is 'First thing for an American' - any ref could have been used her to point towards an American ie; New Yorker or an actual person like a famous President, by using a Pronoun - the setter is pointing straight to an 'American'and thus giving the reference that it's what an American calls 'First thing' which is 'Sun up' (explained in short by jazzgirl above)
Tha second part of the clue is
students' union, apparently
Students union = NUS
and it's going up in the Down clue - so it's 'Sun up'
The two parts are completely seperate but the setter has been very clever in his/her use of Harvard in the wordplay for excellent surface reading and misdirection to the solver.
Hope that all made sense.