1st United States Artillery commanded by Robert Anderson. The're still around today as the 1st Air Defense Artillery.

Major Robert Anderson had just transferred his command—Companies E and H—from Fort Moultrie to Fort Sumter [Around April 1, 1861].

On the 1st of April, 1861 but five posts within the limits of the seceded States were still occupied by United States troops. These were Fort Monroe, Va.; Fort Sumter, S. C.; Fort Taylor, Key West, Fla.; Fort Jefferson, Tortugas, Fla.; and Fort Pickens, Pensacola Harbor, Fla. Of these the four last named were garrisoned wholly or in great part by the First Artillery

Taken from page 306 here: http://www.history.army.mil/books/r&h/R&H-1Art.htm

Now you can start by looking up Muster Rolls for company E & H. I would recommend contacting http://garmuslib.org/ they have a lot of resources including some original and copied muster rolls.