Approximation algorithm for finding the maximum common subgraph in two DAGs
Isn't there a straightforward approximation-preserving reduction from maximum independent set (MIS) in undirected graphs to your problem? 

Given undirected graph G=(V,E), form DAG A=(V,E') by ordering the vertices arbitrarily and directing the edges accordingly, then take B=(V,{}) to be the DAG with the same vertices but no edges.

Any subgraph common to A and B corresponds to an independent set in G, and vice versa, no? 

If this reduction is correct, then your problem is as hard to approximate as MIS; which is to say, forget about it. :-)