Obvious improvments would be to turn off screen updating and calculation while the macro fills in the formulas, then turn back on at the end. There may be additional ways to improve this but I would start with these low-hanging fruit.
sub test()
Dim Total_Rows As Long
Sheets("Sheet1").Activate
'## Disable screenupdating and calculation
Application.ScreenUpdating = False
Application.Calculation = -4135 xlCalculationManual
Total_Rows = 13000
Range("C2", "C" & Total_Rows) = "=SUMIFS('Sheet2'!C:C,'Sheet2'!A:A,'Sheet2!A2)"
'## Put the calculation back to Automatic
Application.Calculation = -4105 'xlCalculationAutomatic
ActiveSheet.Calculate 'Force a calculation
'## Replace the formulas with values only:
Range("C2", "C" & Total_Rows).Value = Range("C2", "C" & Total_Rows).Value
'## Allow the screen to update
Application.ScreenUpdating = True
End Sub