So, as my title suggest, I wonder if the 737NG can still fly without any of it's electric run systems, which means anything that runs on electricity is dead.
TLDR: Yes, the 737 can still fly, the gear can be lowered and one can brake after landing.
A complete failure of every single electrical system is extremely unlikely. The QRH (Quick Reference Handbook) does not even mention this case. Usually, at least the battery remains and can power all emergency systems for at least 30 minutes (60 min, if 2 batteries are installed). See also What happens with the avionics if in midair there is loss of power or fuel flow interrupted?.
Even in the unlikely case of a complete electrical failure, the aircraft is still flyable. As long as hydraulic pressure is still available, the flight controls would function normally. But even without any hydraulic pressure the aircraft can be controlled via manual reversion. This means the mechanical link of the flight controls to the hydraulic actuators at the control surfaces can transmit the forces to the control surfaces (similar to what happens when hydraulic power steering fails in your car). The forces required will be substantially higher than usual, but the aircraft is flyable. Also manual pitch trim via the trim wheel in the cockpit is still possible.
Some things to consider:
A complete electrical failure would include a failure of both EECs (Electronic Engine Control) and therefore both engines would be dead. You would have to sail the aircraft down. As Ralph J pointed out in the comments, the engines could still windmill and might still provide some hydraulic pressure via the engine driven pumps, but far from normal pressure.
Normal flaps extension is powered by hydraulic system B. The FCOM says that the flap lever
selects position of flap control valve, directing hydraulic pressure for flap drive unit position of the LE devices is determined by selecting TE flap position
so this might still work with hydraulic pressure. But note that there are electronic components like the flaps/slats electronics unit (FSEU) involved, which would no longer function without electrical power. I am not entirely sure if that would already prevent normal flap operation. See also the flaps and slats control diagram below.
Flaps can also be extended without hydraulic pressure using the alternate flap extension mechanism, but this is electrically powered, so alternate flap extension would be impossible in this case.
Normal gear extension is powered by hydraulic system A. If the pressure is still sufficient to release the up-locks, the gear can then fall down assisted by gravity and the airflow:
When the LANDING GEAR lever is moved to DN, hydraulic system A pressure is used to release the uplocks. The landing gear extends by hydraulic pressure, gravity and air loads.
Otherwise, manual gear extension is always possible via three mechanically connected cables in the cockpit using a gravity drop of the gear. It is however not guaranteed that the gear will lock in place.
If hydraulic system A pressure is lost, the manual extension system provides another means of landing gear extension. Manual gear releases on the flight deck are used to release uplocks that allow the gear to free–fall to the down and locked position. The forces that pull the gear down are gravity and air loads.
Without hydraulic pressure, both normal and alternate braking is unavailable. The brake accumulator will however still have sufficient pressure in it for braking after landing.
Older 737 NGs still have mechanical standby instruments (artificial horizon, airspeed indicator, altimeter). These could still function. Modern NGs have replaced this with an electric standby instrument. Without any instruments, it will be extremely difficult to control the aircraft.
All radios and the transponder would be inoperative, so no airport would even know that you are coming to land there.
Without any hydraulic pressure, the following components are inoperative:
All flight spoilers inop
Roll rate will be reduced and speedbrakes will not be available in flight.
Trailing edge flaps normal hydraulic system inop
The trailing edge flaps can be operated with the alternate electrical system. Alternate flap extension time to flaps 15 is approximately 2 minutes.
Leading edge flaps and slats normal hydraulic system inop
The leading edge flaps and slats can be extended with standby hydraulic pressure. Once extended, they can not be retracted.
Main landing gear normal hydraulic system inop
Manual gear extension is needed.
Ground spoilers inop
Landing distance will be increased.
Normal and alternate brakes inop
Inboard and outboard brakes have accumulator pressure only. On landing, apply steady brake pressure without modulating the brakes.
Nose wheel steering inop
Do not attempt to taxi the airplane after stopping.
(Boeing 737 NG QRH 13.12 Manual Reversion)
(Boeing 737 NG FCOM 9.20.24 - Flight Controls - System Description)