I asked John Bullard about his banjo setup for classical in this interview. John is the person I think of first when I think classical banjo. He pretty much said it is no different than bluegrass setup and he uses the same banjo for both.
Depends a bit on what you mean by "classical music!" That could mean (a) arrangements of Haydn, Mozart or Gershwin, or of older popular music, or it could mean (b) the banjo compositions of the late 19th/early 20th centuries by Joe Morley, Fred van Epps et al.
Type (b) is usually called "classic banjo" nowadays, and is almost invariably played on a nylon-stringed, open-back 5-string in the old standard tuning gCGBD. Some players use natural skin heads, but most use plastic of some sort. Tastes vary.
For type (a) any 5-string banjo will do. I would, however, recommend the "classic" tuning of gCGBD. This facilitates playing in the popular keys of C, G and F and their relative minors without a capo. (Even C minor and Eb major aren't too difficult!) I play this type of music on all of my banjos, depending on my mood and the auditorium in question. The nylon-stringed open-back is clear and snappy, and not too loud; the zither-banjo with its mix of nylon and steel strings has more volume, and more sustain. My resonator banjo with steel strings and plastic head is the loudest and has the most sustain, but sounds less "cultured" than the zither-banjo.
One point to note is the use of the 5th string. It's used as a stepping-stone from the open position to positions above the 5th fret, or when it's more convenient to thumb it rather than fret a G on the 1st string.
IMHO sustain is important. I'm primarily a guitarist but I play several arrangements of classical tunes (Jesu, Joy of Man's Desiring, Bach's Bouree, etc.) on my Stelling Golden Cross. The Stelling has a lot of sustain, which some have criticized for Bluegrass, but which allows me to capture some of the nuances of the original compositions.
I think the banjo recalls the sound of the harpsichord and lends itself very well to classical music.
I would suggest : Attack/Sustain/Decay. 3 variables.
Snap = Attack Crackle = Sustain Pop = Decay
Now you can adjust your setup, loosen or tighten the head, adjust the tailpiece, change bridges. The Gibson sounds like it might sustain for 60 seconds.
You may wish to tune down 1-1/2 steps, that will put you in E. That will let you explore what a longneck does, it gives you 6 additional keys that you don't get in G tuning: E/A, F/Bb/F#/B. In A, Bb, & B, you use drop C.
Fred Starner got his Whyte Laydie from a pawn shop and took it to a luthier to get a longneck made. The luthier wanted Fred to play classical, so he hid the longneck in a closet, he didn't want him to play that "Seeger crap." Hope this helps.
Not trying to be a wise guy, but I would focus more on the characteristics of the banjo player.
While I feel I can get away with a certain amount of sloppiness in other styles, classical is not one of them. I pay particular attention to execution - attack, proper fretting, articulation, tone and tuning. And unlike other music I play - I try not to make mistakes!