If the people of Middle-Earth made a map of their world or an area of it, did they put west on the top of the map? Or did they have north on the top like the map in the Lord of the Rings looks like? If the former, are there any modern day maps of Middle-Earth that put west on the top?
Almost certainly, yes.
Not always (witness the Dwarven map with east at the top), but it was evidently the usual orientation in the West-lands.
Appendix E to The Lord of the Rings:
The names of the letters most widely known and used were… númen, hyarmen, rómen, formen=west, south, east, north…. These letters commonly indicated the points W, S, E, N even in languages that used quite different terms. They were, in the West-lands, named in this order, beginning with and facing west; hyarmen and formen indeed meant left-hand region and right-hand region (the opposite to the arrangement in many Mannish languages).
Now, this doesn't explicitly say that maps were drawn with west at the top, but "south = left, north = right" very strongly implies it!
Additionally, there is an implied contrast (which Tolkien was probably hinting at with that "Mannish languages" remark) with medieval European maps, which customarily had east at the top (hence the use of "orient", or "east", as a verb meaning "to point towards"). This "facing east" was more than just a language convention, it was a cartographic one too; and so we can infer that the same goes for Middle-earth.