Ok - a bit of clarification, as I originally led the OP to choose this hub about a month ago:
The OP wanted to buy one of these
specific Schwinn Approved hubs to build their singlespeed Varsity. It had to be this hub - but these particular hubs were only offered to fit 5-speed (and possibly 6, not sure) freewheels, and this one is, of course, set up as so, offset to the side to clear the five cogs. Obviously, respacing an old freewheel hub into a singlespeed hub isn't a big deal.
That said:
Petersrock - assuming your Varsity frame is fine, it should measure 120mm. So should your hub. Yes, it helps to get a set of cheap Harbor Freight calipers, and you'll need them for the hub spacing I'm about to explain.
Right now, you have the factory, single large spacer + nut (a one piece deal on these particular hubs) on the right side of the hub. This is what takes up the space to offset the hub on the axle for the 5-speed freewheel which this hub was originally spec'ed with. You want to get this hub much closer to center, thus giving you a better chance at good singlespeed chainline and reducing the spoke offset to the rim.
To do this, you'll want to get some hardware store spacers for
both the left and the right side of the hub. Undo the locknuts and walk the axle offset towards the left side of the hub by turning the hub bearing cones. Once you have it roughly centered, install the new spacers between the outer locknut and the washer for the bearing raceway. You want to make sure there are enough spacers that once the locknuts are cinched down, you wind up with no more or less than 120mm distance from axle nut to axle nut.
You'll also want to make sure the axle doesn't protrude past either of the dropouts when installed after this conversion.
EDIT: Wingnuts? You'll need a replacement solid axle for that.
-Kurt