the first known ascent of the Fineilspitze was done by Franz Senn (the local preast and one of the founders of the german/austrian alpine club) with his guide Cyprian Granbichler and mule drover Josef Gstrein on September 8th 1865.
name: the official name in the austrian map is "Fineilspitze". Old sources have the mountain with "a" as Finailspitze. As a dialect form austrians tend to convert their mountains from feminin (Spitze) to masculin (Spitz) so you can sometimes read "Fineilspitz". But I suggest taking "Fineilspitze" and giving in addition the italian name as Chandra suggested
map: by clicking on http.//www.austrianmap.at you can easily search for the mountain and obtain a detailed map
Best map to buy is Alpenvereinskarte 1:25000 Nr. 30/2 Oetztaler Alpen, Weisskugel, available at DAV lifeAlpin, as linked
Should be FinEil Sp. I had problem to find this mountain. :-(
There is a livecam at the Tisenjoch