Checking in dynamic routing protocols is not diffrent within Cisco ACI compared to normal NX-OS.
If you want to know which LSA you received from your ospf neighbor then check the ospf database. You need to be aware of the ospf topology (area) the ACI border leaf is in. And you need to know for which linksate-adverstisment you are looking for: if the LSA you are looking for is inter (LSA-type 3) or intra (LSA-type 1 or 2) or even from ospf border router (LSA-type 5 or 7). From the ospf database spf path calculation is run and selected routes are put into RIB.
show ip ospf database vrf TENANT:VRF
There you should see which LSA you received from which originating router
What Lukas suggested is the only way to extract the data you need from the switches.
The difference between BGP and OSPF is that the latter exchanges a DB rather than advertise prefixes like the former. Then each node reads the DB information in order to compute the shortest path from its point of view. This means that in order to understand what prefixes are originated from which router you need to read the OSPF DB. Unfortunately this is not an easy task at the beginning but you can find a lot of resources online that could help you with this.