I'm running ~13-year old ultegra 600 group-set on my race bike right now, and it has been working quite well. However, of late, I have noticed that on the right STI, on the inner paddle to shift to lower gears, it doesn't seem to catch and I have to pump the lever several times to get the rear derailleur to shift. Is this problem fixable, and if so, how?
At the suggestion of one of my other team-mates, I sprayed a little bit of T7 in one of the STI hole openings, in hopes that it would free up slippery and clogged up STI mechanisms, but this was not very effective.
I never learned to ride a bike. It is my deepest shame.
It is possible that the tranluscent green grease shimano uses in small mechanisms has hardened up over time - it is notorious for this. If this is the case, you need to flush the mechanism with plenty of degreaser or wd40. This is probably the best starting point as if the shifter is wrecked you can't really make it any worse by soaking with degreaser. If it comes back to life after flushing, carefully blow the internals dry with air and then put a few drops of a light lube (like T7) after.
The other possibility is that one of the ratchet springs that engages the mechanism has slipped off or broken. THis is fixable (difficult, but do-able) if you can access the internals. On my RSX shifter from the same era the left side is open but the right side is tightly protected from repairs... lucky for me only the left shifter has failed.
how does one take this apart? Should I be soaking the entire STI?
No, do not take it apart. Shoot it full of WD40. If you haven't replaced the shift cables and cable housings this could be the problem. Dirt and/or corrosion can cause excess friction within the housings.