You can set up the tank and acclimate him slowly by putting him in a cup with water and every 15 minutes pour out about 1/4 of the water and put fresh water from the new tank in until the cup is filled with mostly clean water from the new tank. It should take 30 mins to an hour if done correctly. However, putting him into fresh clean water shouldn't hurt him at all, just try to make the clean water as close to the same temperature as the old water as you can.
What size is his bowl? I would begin by doing daily water changes with fresh dechlorinated water. Work on getting him into a tank that is at least 2 gallons with a proper heater that will keep the tank temperature around 80*. An adjustable heater is best and I highly recommend investing in a good heater, it may cost a little more than a cheap pre-set one but they generally work better and last longer.
What are you feeding him? I recommend getting a decent pellet food and supplementing with frozen. He probably has a compromised immune system from bad care and clean water and high quality foods will help bring him back to prime condition.
I would avoid heating the tank with a lamp. Its better for him to be on the cooler side than it is for him to have temperature fluctuations between day and night. Instead put his bowl in the warmest part of the house (but not near any windows or vents) until you can get him into a properly heated tank.
Kudos to you for wanting to give him proper care and welcome to the forum
If you have any more questions don't hesitate to ask.