According to my philosophy, yes he should have given you the ftp user & password as well as the directadmin user & password. I always do.
On the other hand, having uninformed users have this information, they could accidently wreck the site or compromise it's security. And worst of all, you might decide you wanted someone else than him to change things.
I don't know your deal with this guy or what you communicated. In any case, it's likely that if you're paying for the server directly that you can mail them and ask them for the user & pass (and ask them to change it first). In that way you, instead of him, have control and the ability to go to someone who helps you better.
An update can break things and the things behind the screen are sometimes quite complex and hard to communicate, because you're not only translating a language, you're translating a culture in a way as well, so please give him a little space for that.
However, any intelligently built wordpress website should be able to withstand updates (emphasize "should", even in complete expert's cases, this does not always work as it "should").
The bottomline is, yes it may take him a long time to fix it, but not if he had worked more intelligently & professionally. I also don't know your previous agreement. I don't know what laws you're under to know if it's time for a lawyer's letter. Where I'm from it's better to take the loss usually and do better pre-business negotiation next time. Again, I don't know your contract.