Yes, you can process your first payroll in January, as you need to. SimpleCity will only include checks dated in the ending year in the Year End file.
However, if it is also your fiscal year end, you will not be able to update your payroll to General Ledger until the General Ledger fiscal year end has been rolled.
Keep in mind that you may need to enter pay raises or make changes to deductions before starting the first payroll of the new year.