ATPL student here - keen to learn as much as possible about the 737!
I thought the A/T is armed as part of the before take off checks, so that the autothrust sets the correct power for take off. Once airborne and vertical path/speed modes are selected ie IAS/VNAV the A/T then controls power/pitch to achieve desired speed/ROC etc.
On finals the A/T is selected off and speed controlled manually. In event of N1 exceeding a preset value (is it about 85%) the auto thrust detects this and sets go around thrust (in the all engines operating condition).