In the first one. Both of them are correct grammatically, but the meaning slightly differs.
I shoot him., suggests a more definite outcome of the shot, that it hits the object (in this case a person, he). While,
I shot at him, says nothing about whether the shot hit or missed, it just tells us that the subject (in this case, I) aimed to shoot at the object and fired.
In the second one, Only
The plane shot missiles at the base. seems to be grammatically correct.
I think that this would be more appropriate. "The plane shot missiles at the weapons bunker on the base." or "The plane bombed the weapons bunker on the base.".