We've surely seen things restored like the Nostromo or the Galactica before.
Aircraft, trains, etc get restored at museums all the time from private donations.
Takes years, lots of volunteers.
My entirely uneducated gut guess......
IF it were privately owned I would say many tens of thousands of dollars and there would be plenty of volunteers capable of the job over time.
Given this is a NASM artifact and all the complexities of that, I would venture costs to be at several hundred thousand dollars, and should bring some more visitors in once complete.
They have a full on Jules Verne display with a mock up of the Columbiad so they do recognize the importance of speculative fiction in a museum pressed for space......
Jules Verne Craft - Image Detail
Just thinking out loud....
Might be worth a public "fan letter" to the Trek community AND to ask people like Tom Hanks, Paul Allen, known big name Trekkies, sure like Stiller if he is big on Trek, to add their names to some kind of public request.
Obviously folks like that have deep pockets, they could cover the cost entirely on their own if they so choosed, but I would prefer that fans also have a chance to prove they will put up some cash and be part of of it.
Again, just thinking out loud, if folks like that are truly Trek types, I can't imagine them not wanting to support a fan birthed restoration request. With the internets seems easier to wave flags and get attention.