How can I update the brake system on this vintage Universal La Riviera ladies bike?
The brake pads are almost certainly hardened and need replacing, and the cables probably rusted and stiff, so also need replacing. Both these items are considered consumable and relatively cheap. 

Before going out and buying new ones, I would remove the cable and check the levers are free. Also check the brake calipers are opening and closing freely and spring back when released.  A spray of WD-40 or penetrating oil may be needed if really stiff, but once freed up a squirt of a light oil or chain lube will provide lubrication.  If you can get these moving then install the new, high quality pads and cables and they should work better than new.  

Replacement brake levers and calipers are easy to find if you're not after exact, era-accurate replacements.  Your local bike co-op or recycle center is likely to have brakes or a donor bike. 

Installing cables is best done with a proper cable-cutting tool. It's possible to cut cables with other tools (such as a Dremel), but it's probably better to hand it over to the LBS if you don't want to purchase a tool and cannot get access to one. 

As suggested by @Daniel, the bike is not worth a lot, and you could very quickly end up spending more on the bike than it is worth. Before spending too much time and money I suggest looking very closely at the tires, as if original these are likely perished and in need of replacement.