Quite often the car parks are on time limits which start from when the car rental is allocated on the computer. Once this has been activated you get something like 20 minutes to find your car and leave the car park before you get charged for parking. This means you can't just book the car, load it up and leave it where it is to go and get food. I would also advise you to wait for Micca before you get the car, because otherwise you'll have to go back through the tunnel again. Think of the tunnel as the 'point of no return' and you're pretty much there!
Which ever way you come east, it's going to be a bit busy. If you're happy driving on the other side, I would suggest you aim for the M4 and come along the North Circular (A406)to the East Ham junction (A117). Keep an ear out for traffic reports - BBC Radio 2 88-91FM is pretty regular with those, or ask at the car rental desk.
If you are not confident just yet about driving on the correct side of the road, then aim for the M25, going north/east, probably signposted towards the M1. Micca should be able to help you there but he's not a driver so his directions aren't always helpful! Keep on the M25, you should be aiming for the M11 junction - it's about 35 miles. From the M11 Junction, aim for Central London or City Airport. From there Micca can guide you and it's the A13 you'll be looking for. If all else fails, keep aiming for City airport. Sooner or later you will come to a bit of road that Micca will recognise