I've designed a timer/driver circuit as per the attached which in theory should meet your needs if you opt for the 12V version of the geared motor you linked to in post #1. This circuit also includes a speed control feature at almost no extra cost. Total cost for parts is only a few $.
If you can't build the circuit from the schematic let me know and I'll try to come up with a strip-board layout (I would also need to know if you wanted speed-control or not).
In the circuit C3 and a 1meg trimpot determine the clocking speed of a CD4060 counter. An LED coupled to the Q4 output of the counter monitors the operation and can be used to set up the timing. The Q14 output goes low every 4 hrs to set a latch comprising two NAND gates U2a, U2b of a CD4093 IC.
When the latch is set, pulses from an oscillator comprising gate U2c drive (via U2d, R10 and transistor buffers Q1, Q2) the gate of a power MOSFET M1 to switch it on and hence energise the motor. The mark/space ratio of the pulses and hence the motor speed are controlled by the setting of a 10k pot.
When the motor shaft output arm reaches the top or bottom position it opens a respective one of two limit switches. This resets the latch and so stops the motor drive pulses.
If speed control is not required the components D4, D5, C9, 10k pot, R10, Q1, Q2 can be omitted.
The attached zip file contains files needed for circuit simulation in LTSpice.