How do you drive from Newark airport to Cape Cod, and where can you stay.

Cross the George Washington Bridge to Rt 95N to Providence RI, then Rt 195 to Rt 25 to the Bourne Bridge. 4 1/2 hours depending upon day and traffic.
Are you asking where to stay on Cape Cod or on the way?

Assuming you mean where to stay on Cape Cod - it's a big island with lots of different villages and hundreds of motels, inn and guest houses. First check out the different villages to see which appeal to you then do a google search for "Places to stay + Name of Village" - you'll find lots of links.

What time of year/day of the week are you planning on? If on a late spring or summer Friday or Saturday you should add several hours to the trip time. Also - before going check the various mapping sites to see if there is any construction going on (to see if you should take an alternate route).

You might enjoy staying at the Commodore Inn/Motel in West Harwich, Cape Cod MA

What kind of accommodations are you looking for? If you plan to visit the glorious Cape Cod National Seashore, Provincetown, Wellfleet, Chatham, and Orleans, then the village of Eastham makes an idea base.