.....Considering that the ram is about 27cm long and 3cm wide and deep, drilling a spindle hole will take some thought; my mill doesn't traverse that high, and my lathe's spindle is too narrow. Maybe a 4-jaw chuck, a thick disc with a 3cm square centre hole, and a steady-rest that can handle the disc..
Can you tilt the head on your mill and use it as a horizontal borer?
If you have to do it on the lathe, maybe it would be safer to spin the drill bit and keep the ram clamped to the cross-slide?
If you can't find any other way, maybe you could make a quick jig. Take a block you can fit in the mill and drill it through. Then pop it on the end of the ram and use a few clamps to clamp it to a 'splint' of angle iron or similar? With that, you could drill it on a drill press, or even just hold the ram with jig on top in the bench vice and if the jig is good, drill it freehand.
And ... if you can't find any way you're happy to drill in to the end, leave the ram as is. Make a block with a pocket in it big enough to take the end of the ram tightly and deep enough to ensure it stays co-axial with the ram, and drill the end of that block as your holder. Then you can have one or more of these as almost Q/C tool holders for the press - you can have a grub-screw or a pip-pin or similar to stop the block dropping off when you raise the ram (or if you want the block to be permanent, dowel it to the ram).
In fact when I needed a temporary bigger foot at the bottom of the ram, I took a block of scrap the size of the pressure pad I need and squared it up. The block was bigger in X and Y than the ram (a couple of inches square IIRC) and a couple of inches tall. I then reduced it in X and Y so it had a square column above a wider base - leaving about a half-inch thick base and the remainder milled down to the same dimensions as the ram with a height of about an inch and a half. I then took a couple of say 3 inch long pieces of angle, drilled through the angle and the foot's column and secured them, then slipped it up over the bottom of the ram and fasted it with a couple of toolmakers clamps.
Dave