No you carn't just plug your cable and your machines in to a switch ...
you can use windows internet sharing..
But a router is by far the better option
I am asuming that your cable connection is though a network card in your computer
you will need to add a new network card to that machine and link that card to your second machine with an x cable
then run Internet Sharing wiz
remember which NIC is Which
If all else fails, You can always try Kicking the daylights out of it, and getting a faster and better system... lol