Well, if you want to make the Viennese original, all you do is combine coffee and sweetened cream, the latter possibly whipped. The coffee can be brewed however you wish, although instant is probably not entirely authentic. :) You can add cinnamon and/or shaved chocolate, if desired.
If you want to make the modern definition of cappuccino, it's based on espresso coffee, which, despite all those labels on coffee packages, is not a particular grind, roast, or variety of coffee bean. Instead, it's a particular brewing method, and one that you're not going to reproduce without the specific equipment. However, there's a pretty close substitute available in the form of instant espresso granules. Unfortunately, that still leaves the steamed milk, which again, you're not going to reproduce without the specialized equipment of a steam wand or a milk frother. You can get sorta-kinda close-ish by scalding milk and then whirring it in a blender at the highest speed you can manage. (A food processor won't work, it has to be a blender.) Careful you don't end up scalding yourself as well as the milk! You can also use whipped cream instead of the steamed milk, but then you might as well call it a Viennese coffee.
The third option is to buy a cappuccino mix - a couple years back there was a brand that advertised itself by having a woman make espresso-machine noises in the kitchen while she simply poured hot water over the mix, after which her friends remarked, "We didn't know you have an espresso machine". I don't know how those mixes get the foam to develop, and I probably don't want to know.