The power rating on an adapter is the maximum power that it can deliver. If you have a 12V 1A adapter, it can be used in this situation. I CAN deliver more power if needed, but WILL deliver the 0.5A that the router requires since that is what the router will request. Think of it this way: In your house, each circuit CAN deliver either 15A or 20A depending upon the breaker for that circuit. When you turn on a TV, does it use 15A or 20A, NO. It uses the power it requires only. Similarly here, the router will only use what power it requires, so using an adapter than can deliver more is OK.
In regards to the different voltage rating: That will depend upon the router itself. Many devices have a voltage regulator as one of the first components on the power line to regulate the input voltage to what the device wants. If this is the case with the router, using a 13V adapter should not be a problem. In general however, I would not recommend using a different voltage adapter. It is too risky. Higher current capability is OK, higher voltage not OK.
Dan