This is a broad question. You should get a a bunch of good answers with a variety of suggestions. In the meantime, I hope this general info helps:
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Replacing the heater core on pretty much any Mustang since around 1991 is generally considered one of the two most detestable jobs that needs to be performed. The other being a complete timing set replacement.
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Replacing the heater core essentially involves draining the entire coolant system
and AC refrigerant system, because they are intertwined, disconnecting all the hoses in the engine bay, then removing the entire dash, center console, radio, steering wheel and all the associated components. It is time-consuming and tedious and is best done by someone who has actually done it before or by a skilled tech who has access to Ford technical procedures and knows what they are doing.
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At any competent shop or dealer, the time you will be charged to remove everything to gain access to the heater core will be 4-5 hrs of shop labor. Minimum. It is just an awful job and a lot can go wrong depending on the level of corrosion and vehicle condition. This is what it will look like.
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At this point, there are two primary differences as to which way to move forward.
1) Ford's standard procedure is to replace the entire assembly that sits inside the car, behind the firewall, on top of the transmission tunnel, just below/behind the dash, in front of the shifter. This whole assembly contains the heater core itself
and the AC evaporator core plus other internal components. It is about 25 lbs and looks like this. Ford considers this entire assembly a sealed unit and "not serviceable". This is about a $500 module online.
2) Despite Ford's stated procedure, the above unit can indeed be taken apart to replace only the actual heater core inside. Note the heater core itself is only about $45 and is shown below. That being said, disassembling that entire unit requires drilling about 15-20 plastic rivets, keeping track of internal components such as O-rings, and then putting it all back together - correctly - with / or without replacing the AC evaporator core which is also inside. Hardcore enthusiast DIY'ers will take this approach. Shady shops will tell you / sell you the entire assembly for $500 and yet only replace the $45-50 heater core. You need to understand which approach they will take. This is where shop / dealer integirty and honesty enter the discussion.
Given the overall invasiveness of the entire procedure, and the possibility of something going wrong, I would not recommend just hacking it apart to replace the heater core. I would replace the entire unit. This is
not a job you ever want to repeat.
3) Re-assembly of everything, in reverse order of above, is another 4-5 hrs of labor.
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All In Costs at a Good Shop or Dealer
8-10 hours of labor and $500 in parts for a job done correctly
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If you can do it yourself, you can save a $$$$ of money.