|Clearly you have planned ahead with the piping. A one foot distance is more than enough. Dirt is one of the best insulators and radiation [electronic signals] blocking substances in the world.|
You will have no problem pulling a coax cable to the outbuilding.
Problem becomes splitting out the internet signal from the tv signal.
Usually you have a split in the house for cable and tv. You have to have your cable provider supply the equipment for the outbuilding which incures extra cost.
Now on the otherhand you could run a coax AND a cat6 [why put in 5 these days -cost is minimal in comparison]run through the pipe.
Depending on if you only planned on one pc or multiple network devices would decide if you wanted to wire it crossover to straight thru.
You can order premade any lenght cables off the internet.
Reason I mention this is if you go with the ethernet cable from the router to a switch that doesn't have an uplink port you need to wire crossover.
If just to a pc or to a switch that supports mdi/mdix [auto sense auto correct if xover or straight] /or has an uplink port you want to wire straight thru.
Google cat5/6 wiring and look at the diagrams and how tos. Easy to do if you have any mechanical/construction aptitude. Otherwise order it :-)