You can use a board, or just a piece of paper and mark down where you are. A lot of people use a big plastic mat that you can draw on with grease pencil and then erase. Some people use a little lead figure 1 inch high and move it around, or you can use any little thing like monopoly. A lot of PnP adventures does not need a map, the DM just talks and explains what the characters see, and then the players say what they want to do. It can be sort of casual and unspecific or very detailed depending on the style of the DM and players.
It's hard to have a big battle like you have in Shadows of Amn in pnp because it takes forever to figure out everything and roll all the dice.
Computers are better at staging a big battle because they keep track of everything and make computations quickly, but people are better for storytelling and imagining an adventure. For instance, if Shadows of Amn was a tabletop game, a battle like you have with those bandits in the sewers could take hours to play, but on the other hand in the story you would have more freedom - you could ask the DM all kinds of questions and do things that the game won't let you do. You could get more involved in the politics between the temples and the thieve's guild, or the Radiant Heart, for instance.
Whatever part of the city interested you most could be expanded as you go, and the DM would be able to stop you from exploiting things - He or She would probably not let you get away with having 200,000 gold, or stealing back too much stuff. If you tried cheesy cloudkill on a dragon, the DM would say, "the dragon blows a big flame burst at you and the cloudkill goes right in your face", and if you were killing everything too easy the DM might send some crazy bounty hunter after you.

Of course, you would never be able to reload the game or play the same adventure twice, so you wouldn't know where everything is. But then the DM can have compassion on you too. If you were dying in a battle, or did something really stupid, the DM might have help show up, or have the floor cave in and everyone gets washed away in a flood, or anything...
Anyway it all depends on the DM. PnP can be a lot of fun if you find people that you get along with to play...It's more like a party - get snacks and drinks and hang out with your friends.