Start by picking up the newspaper on the bench. That will let you stay hidden as she walks past. She'll turn around at the end of the parking lot, so get behind a car and wait for her to reach the side walk. Stick close to the fence in case you need to hide, then run up to the shoe shine stand and let her get a fair distance away from you.
If you stick about a block away, then she shouldn't ever look back. Wait for her to cross the street and enter the neighborhood. You can then just move from cover to cover and you'll be fine. Once you hit the dumpster, you'll get a checkpoint. Then just run from point to point, using corners, cars and cover until you see her go down the alley. Then just follow her around to the bookmaker.
Go inside to learn about the note she took. Then just go to the telephone and shade in the notepad on it to reveal the address of the Examiner Drugstore.
Take the trip over to it and then go inside to find that she was there too. Look at the card to get the cab company, then use the phone to put a trace out on her cab. Mervin will also give you the address for Ray's. Drive over to Ray's to catch her as she leaves in the cab.
Tail the cab as usual, until it goes to the bus depot. If you can avoid hitting anything and stay about a block or so behind it, you'll be fine. Wait for her to enter the depot, and then move inside yourself. Take a seat with the newspaper, and just watch her until she goes into the bathroom. Walk toward the bathroom, and then kick the door in when you hear the shot. It seems like things went in an unexpected turn.
Once you regain control, just look at the ground. Pick up the revolver and pop open the chamber to confirm that one shot was fired. Then open the handbag to get the other half of the ticket and the only lead you have left.