It appears that this engine has a blind cylinder so reaming will not work as reamers have a leade on the end to aid insertion .
If i were making this engine i would use the 19/64 drill to remove the bulk of the material and only drill deep enough so the point of the drill reaches to full depth then use a 5/16 endmill in the tailstock chuck to clean up to size and square the end of the cylinder up to full depth.
Then make the piston to fit .
A much more rigid set up for the lathe is to mount the cylinder up in the toolpost or to an angle plate fixed to cross slide and put the drill / milling cutter in the chuck but may be more work than necessary for this engine.
You could of course do the same operations on a mill or even a drill press .
I would not recommend that a beginner tries to bore a blind hole let alone one so small , much better to learn on through holes first and preferably a larger diameter .
The problem with really small boring bars is flex and they will flex causing a tapered bore , not hard to fix but you have to account for it .
Ian.