[RUNNERBUN]

The short answer is yes, you can do 350 miles in a day and I have. But! do you really need to or want to. I have done many thousands of miles on long trips and usually work on an average of 220 to 230 miles per day. This gives you plenty of time to enjoy the sceneary and time to smell the Roses.
As others have pointed out, your age and health are major influencying factors. Rest stops are very important for your own safety as well as your passengers and other road users.
Your travelling speed should be one that both you and your car are comfortable with.
Your average speed is not so important as is the lenght of stops you make along the way. You will find that what should be a quick fuel stop can easily take half an hour or often more. Lose ten minutes and then try to make it up by speeding, you'll find that is nearly impossible. Our cars attract a lot of attention and you will find that a fuel stop or coffee break will often turn into an oportunity to meet many interesting and friendly people. (a part of our hobby that I particulary enjoy)
As this is your first long trip I would advise you to make it a 2 day affair and enjoy it. Should you then consider it appropriate to do the distace in one day maybe you can try it on the return journey home.