Go take a look at the Citizen eco drive range.
They all use a solar powered quartz movement that is pretty much worry free sand one of the most reliable options out there, and come in a wide range of styles within your budget. Pretty much my default recommendation for a simple, no non-serviceable watch on a budget.
Failing that you are looking at options like the Seiko ranges (again, they do a range of solar quartz movements that should fit your budget), and various Casio and Timex models.