One of our forum members, @gravling, is an absolute master of keeping her Heat low in the NBZ. Heat management is a big part of being successful on the dangerous streets of the old city, and here is some concise and awesome advice straight from the Expert herself!
1. You cannot reduce heat by sleeping at the same hotel you slept at last night, or the night before, or the night before that. You need to move around.
2. Stealth exit instead of normal exit your battles.
3. Buy a taser or stunna, and just knock your targets unconscious instead of killing them. The neighbourhoods take it a lot better.
4. If the heat gets bad -- particularly after 'the spy attack burns you' head for a hotel. If you keep walking around you will keep attracting enemies, and the more you fight them the more your heat grows. Its Death Spiral time.
5. If you have a good rep in a neighbourhood you won't notice the heat -- or notice it as much -- there. Thus you need to build up your rep with some faction or 3. The secret to getting a good rep is to do more good for them between sleeps than burns off every night.
The more good you do for them, the longer it will take for it to burn off. If they only slightly like you, they will be quick to forget you. If they like you a lot, then each bar of liking will endure longer -- the relationship is not linear. But it is very hard to do enough jobs to get liked that way when you have only a 1 job computer. Upgrade to a 2 job one as soon as you can.
Try to find work that gives you 2, and then later 3 jobs in the same neighbourhood. If you spend all your time travelling from job to job you will never build up the necessary rep.
6. Don't be afraid of cancelling jobs. I.e., go to the Yakuza Gambling Den with a 2 job computer. Take a job. It will either be to Los V (in Roxbury Crossing) or somewhere in Fennian land. Then take another. If they don't match, cancel one and keep drawing another job until you get a pair. You will gain rep by not travelling all over that offsets what you lose if you pick a Connector who has limited choices in where he sends people.
7. Walk instead of using taxis unless it takes you more than 2 hours to walk there. All taxi rides take the same amount of time -- and the thopters and the water taxis take no time. If you are spending all your time in taxis you will never improve your rep enough for it to matter.