Terrific find. Envious!
Aside from replacing the tires (and maybe the brake pads, if they've hardened to the point of being useless for stopping at speed), you might want to consider modifying the gearing by installing a larger sprocket on the rear hub. (That's definitely a sprinting gear, appropriate for velodromes but too high for even moderately rolling terrain.)
You'll probably need both a larger sprocket and a new chain, since that chain looks as if it won't accommodate an appropriate sprocket (e.g., around 18 to 20 teeth, since that appears to be an approximately 50- or 51-tooth chainring).
Also, to get used to the fixed gear, try deliberately "coasting" every 30 seconds or so throughout the first few rides or until the new technique of easing off on the pedals rather than stopping pedaling begins to feel familiar. It might help to keep practicing the same technique on your other bikes, too, so that the habit becomes ingrained.
Last edited by Trakhak; 12-22-20 at 06:16 AM.