Will Bella Cullen stay the same age forever?

That depends on how you define age... Every year, it will be another year since she was born, but, like Edward and the others, she will not appear to age, suffer the effects of aging, or visibly grow older at any perceptible rate.

Or, to borrow a phrase, she'll be 18 years old "for a while". :-)

When you consider the Volturi and other very ancient vampires in the books, they do show signs of aging, but they are very, very old... Perhaps thousands of years.