If its a Racehorse to Riding horse qualifier, the judge will ride. It will be as per a riding horse or hack class. The standard is usually very high, it qualifies for a final at Hickstead and that was last year mainly attended by pros.
Here is a judge riding my horse in a retraining of racehorse class:
And to clarify hack or riding horse turnout - Mehitabel has an excellent website here:
http://rehorses.com/
But here is me in hack turnout:
And again in a riding horse turnout:
I think that would personally be a bit much for first party, you would be much better off having a fun small outing and do some unaff classes like Best rider, best horse etc.
As for tack - for local and infrequent showing you can wear anything as long as its very very clean!
But if you are going be serious about showing, and do the ROR/SEIB qualifiers etc, you will need to go the whole hog. Brown straight cut show saddle, brown hunter bridle with double bridle. LIke I say, if you are going to compete in affiliated classes you DO need to look the part.
Edit to add to compete in ex racer classes you need a WHITE Weatherbys passport where the front page is called 'Document of Description' - does yours look like that or is it a standard equine passport that happens to be from weatherbys? If you can prove his racing name (what was it , out of interest?) and get some bloods done, you can get his old racing documents back to entitle you to do these classes.