No need to rush - you’ll get a better juice yield if the apples sit in buckets for a couple weeks or so before you press them. When the surface looks a little shrively, that’s a good sign.
I’ve seen people use a screw jack with a round wooden foot to squish the apples in a bucket. I suppose you could rig up a lever with a plunger going into the barrel, and then just suspend weights from the end of the lever. Kind of like an old school bottle capper.
The “foot” on my press is a round piece made of 2” thick maple boards laminated together, with a cross-piece that goes the opposite direction of the other boards, and the screw jack presses on the cross piece. Not sure if that makes sense? I’d take a picture but it’s raining.