Round as like a ball---or round as like a table edge?

Round like a table edge. Actually it's a handle on a chest of drawers, about 12in long with approx. 3in radius on the front side.

Many methods, just about any way you can do veneer:
- Hammered with hot hide glue
- Vacuum press
- Curved cauls and clamp
- Yellow glue, allowed to tack, then ironed on
- Hot-melt banding, ironed on
- PSA self-adhesive, peel & stick
- Contact Cement (assuming paper-backed veneer)

So, the surface is curved---is it a flat curved surface or a rounded curved surface?

The flat curve is simple. Use an iron on veneer or a contact adhesive, easiest since you probably do not have the equipment/tools for the other methods.

The rounded(no sharp edges and the veneer needing to bend around the curve and the top/bottom of the curve) will be darn near impossible.

Would be easier to make a new handle.

I think Handymac means to ask whether it's a simple or compound curve. A simple curve should be fairly easy. A compound curve, though -- not so much.