Reklaw - I just did my Jirachi wondercard this evening! ^^ It was actually pretty simple - if you go to Smogon's RNG guide there's a page for wondercard Pokemon. Ignore most of it if you already know how to RNG on HGSS, the important bit is the seeds/spreads in the table. Pick a spread you like - natures don't matter, they're assigned randomly - and save in front of the delivery guy in Cerulean. (no NPCs!) From here just RNG as you normally would with your chosen seed from the table, and just repeat the process til you get a nature you like.
PS I don't know if it was just my game or a general Rachi wondercard thing, but when advancing my frame I had to stop on the frame after rather than the frame before my target frame. Dunno if it'd be the same on yours, but if something goes wrong and you don't know what it could be that. Good luck!