I have a new computer downstairs with DSL internet connection through my phone.I would like to take the old one & hook it up upstairs to use occasionally. I do not want to network the computers, what do I need besides a router & cables. CAn I use the router from downstairs or is there some special thing on my downstairs jack I will need as well.

You don't need any additional equipment except make sure you have all of your dsl filters in place and that the computer upstairs can reach the router by wire or wireless.