Beyond the usual second hand checks (does it work, are the screw heads buggered, can I see a bloody OX Spring inside, has some genius carved his initials in the stock with a blunt penknife?):

Check the pistol grip isn't starting to develop a small crack.

Check that the safety works. They often don't (bad design).

Check breech lock-up. In particular the wedge on the receiver that engages the ball bearing on the barrel housing. If it is worn, you will string shots vertically. Fixing it is easy if you have the factory replacement parts. But you can't get them anywhere. Fixing without them involves weld, and skilled metalwork. Which is cheap and easy if you have the skills and kit, or expensive and annoying if (like me) you don't.

Check the trigger pull. Sports seem to be different in trigger quality, ranging from quite poor to very good. Improving poor ones takes a bit of knowledgeable smithing, not just twiddling the adjustment screw.

Sights, as said before me, are v hard to find and pricey when you do. The 124 and 127 sights are different, with the longest range markings being 50 and 40 metres/yards respectively. But do you need them? These beg to be scoped. Sportsmatch still make a dedicated mount for the FWB (and Webley Omega) that has a transverse pin to engage the cut-outs in the scope rail. It is sacrilege to put any scope on that isn't a 1980s Japanese-made gloss one, no larger than 3-9x40.

With the best contemporary brand of pellet (eg proper Wasps, Superdomes, H&N Silouhette) that suited a particular gun, FWBs regularly (in either calibre) shot into 0.30" at 25 yards in the 80s, and would usually shoot a variety of the better brands all into half an inch or less. They will do as well or even better with modern JSB, AA, and other quality makes.

Until the HW77 arrived, the FWB was the top FT rifle, despite its lighter weight, potentially greater hold sensitivity, and worse trigger than the HW break-barrels. (Because it's a better rifle than the HWs.... Ducks and runs for cover. )

A good Sport is a marvellous thing.