Online/approximate weighted and capacitated bipartite matching
I think there is no paper solving that exact problem, but "Online Vertex-Weighted Matching" by Aggarwal, Goel, Karande, and Mehta (2011) is very close. If I understood correctly, they solve your problem only with all capacities equal to one.

My best guess is that you will have to do some work to extend their guarantees and algorithm to your setting. On the other hand, a greedy algorithm probably should guarantee 0.5 of optimal by a standard argument: Consider the opt matching; for every "slot" in OPT left unmatched by greedy, there was some other match made that was more valuable (since the slot was available, its partner must have gone elsewhere). I hope you will forgive this discussion of methods contrary to your request.

(edit: I want to mention that this argument wouldn't work if the weights are on the arriving vertices; it's good to think about why.)