so if you go to https://jmapb.github.io/nycaabs/?search=1089808 it'll load all the building info and in the table below you'll see the first section called "Footprint". There you'll see the date this (version of) building was completed, if it's completed/demolished/under construction, and its height. The newest height is most likely the current one..
You can also click on "View BBL @ ZoLa" to open Zola and there you find how many floors a building it, how many apartments it has (and other unrelated to osm, but still interesting info).

jmapb's site also links to zoning documents. Zoning diagrams type "ZD1" most often have architectural drawing of the building with dimensions and heights of all/most sections of the building. They are very useful if you ever get into detailed 3d mapping of buildings.