Minister / Jawfin
I had that very issue at Downpatrick some years ago when discussing livery policy in general. Thankfully, with very few exceptions it is DCDR's policy to have accurate liveries on things; if you broaden the net wide you often find that other museums don't.
There is nothing to see on No. 33 which indicates what livery it had. Certainly, it's later life was in GNR ownership, so it would have spent the greater part of it's working life in GNR polished wood / brown, and maybe that's appropriate given that it is now in ex-GNR condition, as opposed to its original condition. There is some evidence to suggest that it was altered somewhat during its life.
The interior is likely to have been like the 1873 Isle of Man "Director's Saloon" which is now in Port Erin Museum. This vehicle started life as a four wheeler but was placed on a bogie underframe along with another four wheeled saloon to make the special coach for the island's Governor through the years. It will certainly have had this layout - that is, seats round the edges sitting against the sides, not facing direction of travel.
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I remain unsure of where my predecessors in DCDR managed to trace the number... I do seem to recall that is was taken to be No. 30 at one stage, though I suppose it may not matter now.
In its original state, I can't help feeling it may have been either a dark blue (between navy and royal blue) or a dark maroon. There is a very old coach in the NRM which appears to be a shade of green which is almost black. As to lining and lettering, leave this with me - I have one resource which might give some clues if I go through it in detail, but with no index this will take time.
Some early coaches, as I mentioned before, were "colour-coded"for the benefit of the "great unwashed"! But a coach such as this would have been reserved for the use of the "gentry". Occasionally in those days, the livery of such vehicles followed contemporary stage coach design, thus deep maroons or even black lined in gold.
Having just done a good bit of delving in that other-worldly chaos masquerading as my study, I find nothing beyond the above on Ulster Railway liveries. However, I have one further idea; if it produces anything I will post here.