I'll echo the suggestion for PHB3 Hybrid rules- it is the closest thing you'll find to 3.5Ed's multiclassing. You'll be more comfy with it.
That said, one of the nifty things about 4Ed is that its various multiclassing and quasi-multiclassing options- by feat, by Hybrid, by class (bard) and by race (Half-Elf)- all more or less work together. For example, a Half-Elf could take a Hybrid class combo AND multiclass feat.
Sticking with your dimension-hopping origins, Human seems most likely...though you could make a case for the trip having reactivated some kind of dormant magical element your character had in his ancestry (because magic in our world is dead), hinted at in old family legends about his "witchy" forefathers. With that possible twist in mind, this opens up Half-Elf, Half-Orc, Changelings, Tieflings, Shifters, Devas as possible options.
For class: Artificer, Fighter, Hexblade, Paladin, Psion, Ranger, Rogue, Sorcerer, Swordmage, Warlock, Wizard and Warlord all have some merit for a PC who started in our world.
Fighter, Ranger, Rogue, and Warlord are, in some way, the most mundane of options. Ignoring the "wahoo" elements that creep in at higher levels, they're the ones that most closely resemble "mundane" classes, and represent the least amount of change from the RW to a world of magic.
To me, Hexblades, Paladins, Psions, Sorcerers, Swordmages and Warlocks could represent the next step- classes in which either your PC has discovered an inner wellspring of power or has made a pact with some greater power to gain unearthly abilities in his transit from one world to the next.
Artificers and Wizards are generally people who have mastered arcane knowledge of the fantasy setting, though again, it could be knowledge gained from reading books in THIS world that just happens to work in the Fantasy setting.
Your character concept reminded me of an old, old character named
Murlynd. In a similar vein, perhaps some of your 1500's technology became magic as well. The gunpowder weapons of home could become the eldritch blasts of a Warlock. Or the creations of an Artificer.