The RF modulator has three sets of jacks:
Coax input (from the wall)
Coax output (to the TV)
Composite/S-Video/Analog audio input (from a DVD player, video game system, etc)
The Slingbox M1 isn't getting video because the input of the Slingbox is attached to the input on the RF modulator--there's no video signal. The only output on the RF modulator is coax. The Slingbox M1 (and all recent-generation Slingboxes) need a device to tune and decode television (typically this is a cable or satellite "box"). If a regular "cable-ready" VCR works with your apartment's cable system, obtain a VCR to tune the channels and subsequently feed the Slingbox. If the VCR works, you'd have these two options for wiring:
TV and Slingbox show only the exact same TV program
Wall -coax-> VCR -composite video (yellow) & analog audio (red/white)-> Slingbox M1 -composite video (yellow) & analog audio (red/white)-> TV
TV and Slingbox can view different programs simultaneously (note there is only a single coax splitter below, but it's feeding two different devices):
Wall -coax-> coax splitter -coax-> VCR -composite video (yellow) & analog audio (red/white)-> Slingbox M1
coax splitter -coax-> TV
An example coax splitter is available from Amazon here. You may wish to investigate options for using a cable box in your apartment--a cable box outputting HD will help the picture quality provided by the Slingbox.
One other note--the Slingbox cannot change video formats--in your current wiring setup you have composite video (yellow RCA cable) input for the Slingbox and component video (red/green/blue RCA cables) output. Ignoring the current difficulties with your apartment's cable system, this will not work--the Slingbox must feed the same signal through the "out" jacks that was fed in through the "in" jacks.