If the temperature outside is cooler than 40 degrees(the stronger the wind, the better!), you could let that deer sit for 24 hours and still use the meat without any problem. Should the outside temperature be higher than 40 you want to find and get the deer cooled below 40 degrees within 4 hours. After 4 hours the potential for bacterial growth is too high to consider consuming the meat.
Gut shotting a deer makes no difference whether or not the meat will be good. Simply take a damp towel and wipe away any mess and wipe with a dry towel after that.