Finger joint pain is typical of RA which is why they will be testing for it. The tablets should provide some benefit pretty quickly (a few hours) with, I think, a slight cumulative effect as well. If they aren't providing you with any relief go back to the Doctors and ask either to adjust the dose or try a different anti-inflammatory. What dose are you on BTW?
PPIs are Proton Pump Inhibitors. Naproxen is an NSAID (Non-steroidal anti-inflammatory drug), which puts it in the same family as Ibuprofen (albeit atronger). NSAIDs are effective, but associated with stomach upsets and potentially ulcers. PPIs work by reducing the amount of stomach acid that you produce, thereby reducing the risk of side effects such as stomach upsets or ulcers.
Whenever you take NSAIDs you should be prescribed PPIs as a preventative measure to protect you from the potential side effects. I would suggest raising this with your Doctor next time you see them.
If they are doing the first set of blood tests then I'd expect they will be testing for Rheumatoid Factor, which is an indicator of what might be happening.
If the RF comes back positive it doesn't necessarily mean you have RA, as it can be elevated for a number of reasons, but it is an indicator (along with other symptoms). Similarly if it comes back negative, it doesn't definitely mean you do not have RA, so do not let things drop if the pain continues.
I'd also expect they will test for ESR and/or CRP, which are indicators of inflammation. If these levels are raised then, again, this is indicative that there is something going on which requires further investigation.
I'm surprised the tests take three weeks to come back - they normally get turned around within a few days. Give them a chase!
Sorry - very long post! Let me know if you have any more queries.