Go here to get manual for your SP-25: http://www.vinylengine.com/library/garrard/sp-25.shtml
I believe that your unit is afflicted with Garrard Gorilla Snot, which is what that old English grease becomes after 45 or 50 years exposure to air and other stuff. The old lubricant can freeze all sorts of mechanical apparatus that should move freely tighter than a drum.
Take a look at diagram 11 on Page 11 of the manual, and you will see how to remove the C-clamp at center of turntable. Once you do this, you should be able to lift turntable off. Now take stylus(needle) out of cartridge. This will keep it from being damaged while you work on unit. Now pick the arm up and move it slowly and gently across the turntable until you reach the hangup point. Arm should not hang up until it gets within about an inch and a quarter from spindle. If it hangs up in an area where a record would have playing surface, you need to see what is interfering.
Make sure that trip pawl on the geared wheel under the turntable moves VERY easily. It might be the cause of your problem. You will see that pawl on the page that shows the picture with turntable removed. On Page 14 it shows the areas that should be lubricated with fresh grease after you clean off the old residue. I use automotive wheel bearing grease for this application becasue it stays put and is pretty slippery, but some like that spray-on white lithium grease. I have used both.
You might find that arm has rotated out of position, but this is rare, unless the arm has been twisted forcibly. I believe that there is a clamp on the underside of arm, and you may be able to loosen it and rotate arm back to proper position.
And good luck from that slippery (some might say snotty) old greaser,