[thirtiesradio]

If you have the early arch top you won't need to take it apart as you use one piece from one side bottom, across the top, to the other side bottom. At least this is the "cathedral" cabinet style I was thinking you have. Removing the side bottom scribe mouldings is necessary but there is no need to take the cabinet apart. You may be having trouble if trying to apply non backed veneer as it won't easily hold a bend. You need the thinner paper-backed type. I wouldn't use wood glue on non-backed veneer as it can bleed through the wood and keep it from later receiving a good finishing. Use contact cement for paper backed veneer and veneer cement for the thicker non backed veneer. I use a special glue pot and the "blue" contact cement for the thinner paper back types.
If you have the later "peaked top" style I'd also use the paper backed veneer as it is much easier to bend. Use the paper backed thin veneer on either cabinet style because using the "correct original" non paper backed veneer requires extensive equipment to get perfect results. With non backed veneer you use a veneer softner and put any flat pieces like the side pieces in a vacuum press to flatten them out completely. This won't be necessary with the thinner paper backed veneer. I use a veneer press for replacing veneer on shapes like the top but since you are doing it without a press I'd use the thinner paper back veneer and contact cement. I use a virutex trimmer for with-the-grain cuts and a veneer saw for cross grain. If doing the front you would also need a belt driven trimmer for the grill cutouts. Way too many tools for just one cabinet.

With minimum tools on either cabinet you can get by with paper-backed veneer, sandpaper, contact cement a roller and razor knife.