They're very nice cars but being a Fleetwood Brougham, the car came with a raft of standard accessories that can dramatically increase reconditioning costs, should repairs be necessary. One of the most common problem areas are the front and rear power vent windows- whose gears are easily inclined to strip. Replacements are available but that's in addition to the regular side windows for a total of eight. The other biggie is the air conditioning system. 1965 is the second year for the Automatic Climate Control system and although improved somewhat over the 1964 version, can still pose a challenge sorting out if problems are present. (Other members better versed on CC systems may have more to add) Fleetwood door panels had walnut veneer inserts which often need refinishing. Finding replacement parts for these as well as all other Fleetwood-specific trim is much more difficult being lower production cars.
This is only scratching the surface.
In short, this is not a car you should be considering if in need of any type of major restoration work. The cost of doing so will exceed the final value by a factor in the multiples. $4,500 sounds fairly cheap but in all honesty it's likely to need a lot of work which is why it's priced at $4,500. It would be extremely helpful if you could post some photos and I'll be able to render a better analysis. The very best advice would be to suggest that you search for the very best & finest example to begin with. Even then, you're likely to spend a minimum of $3,000-4,000 just in basic mechanical freshening and other assorted odds and ends- and that does not include restoration work. I would far rather buy the "right" 65/66 Fleetwood for $15K than the wrong one for $1,000. Believe me, it will be far cheaper in the long run and the car will be far better than the $1,000 car can ever hope to be.
Just one man's opinion.