Ok- - - - -the part we're concerned with is the pushrod seat at the far right of your photo- - - -and I can't see the top of it. Lifters that oil the top end with hollow pushrods have a tiny hole that bleeds a little oil through the lifter and in the process, purges trapped air. If the hole isn't there on yours, I'd suggest obtaining a very small solid carbide drill bit, maybe 1/32" or so, and drill the hole, using some type of home-brewed fixture for the lathe to hold the pushrod seat in a 3-jaw chuck, and drilling with a tailstock-mounted Jacobs chuck. I have a 1/4" capacity chuck mounted on a 3/8" straight shank that I use for tiny drill bits, that I hold in a bigger chuck with a #2 Morse taper in the tailstock. That's how I make custom-size carburetor jets when I'm tinkering with mixtures. It would also be possible to mount a Dremel tool chuck that way.
Jerry