No you do not level up offline, Leveling up in multiplayer unlocks weapons, mounts (I.E. - Horses to ride) and different titles you can display above your name. If you want a lot of detailed info on multiplayer including what levels unlock stuff you can go here: http://reddead.wikia.com/wiki/Redemption_Multiplayer
You will level up and be solo... unless you specifically invite someone of course. --- Feel'n splendiful... "This stuff will make you a god damned sexual Tyrannosaurus, just like me." DA F************* ???