How to check citations and references match in a manually prepared document with APA author (year) citations?
In the "old days" this was what had to be done manually. I simply printed the reference list (one-sided print) and went through the text and checked the references I passed in the text and checked the corresponding one in the reference list. This would take maybe 20-30 minutes(?). It is clear that this is easier to do on paper than on screen but it is a very safe way to do the checking. In the end you will (hopefully not) end up with references missing ticks in either the text or the list. 

So considering, the time it takes, doing the manual check on paper copies, is perhaps boring but not a terrible loss. Sometimes you pick up on other errors as well. Although one should not rely on external help to solve ones own problems, many journals use copy-editors that check for inconsistencies so there may be a back-up for the stray miss.

As for software, I cannot point at one directly. It would have to be able to match first author name and the year between text and the reference list. I does not sound like an impossible task but I would probably just do the job the manual way in the few cases where this is necessary. You can also ask your collaborator to do a separate initial check (as "punishment" for not using bibliographic software). In addition, if you author in LaTeX then writing the bibliography directly with \bibitem and using \cite (or natbib cite commands) commands will at least do half the job.