Remove the screw and the old needle, then put in the new one in the same orientation (needle point down) and center it, then snug up the screw (not too tight, just enough to hold).

Looks like a Pfanstiehl 700 series would work, which can cross-reference into other brands. Presuming you’d be playing old stereo and mono 33 and 45 records, I’d suggest a D7 which will last longer than an S7. If you also want to play 78s then you’d either need to swap out to a D3 or S3, or you could go with a S2 and use it on 33, 45 and 78, but that “jack of all trades and master of nothing” won’t sound as good on anything, and can over time put improper wear on the record grooves (and the D2 would be harsher).