Won't be the transformer. I suggest you read the Williams SYS 3 - 6 Guide to familiarise yourself with the game. Certainly, you should not go changing anything before understanding exactly what you are working with.

http://users.esc.net.au/~boots/Pinball_Guides/default.html

You will find the manual with schematics here

http://www.ipdb.org/machine.cgi?gid=1062

It definitely COULD be the transformer, I wouldn't rule it out you never know.
I had the same exact problem with my uncles Trident.
Wax was dripped around the transformer and it was stuffed, only difference was when I connected it to the mains it would always blow the fuse.
I even tried wiring it for 100V and running a 240v to 100v transformer and it just sizzled.
In the end I stripped back some of the layers of paper and sure enough it was cooked.

You could try to test the primary windings with your multimeter (without the power on) to see if there is continuity, but you may have to disconnect it from the machine.