Water seeks it's own level so it could concievably run up to the elevation of the level of water in the barrel, but not higher than that.
Water doesn't naturally flow uphill so if you want your water to go to an area higher than the barrel, you will need to pump it in some way. How large a pump or the method is dependent on how much water you need to move and the rise to where you want it to run.
A small electric pump out of a camper would probably suffice for moving enough water to flood irrigate your garden, if you want to sprinkle you will need a larger pump with more pressure.
Installing a spiggot in a barrel isn't hard, just get a hose bib and some latex sealant from any hardware store, drill a hole in your barrel, put on a layer of sealant, screw the bib to the outer wall of the barrel, let the sealant temper for a few hours and you should be set to go.
Good Luck