it would take a good rad shop maybe two hours. SB is spot on, it is well worth doing. Otherwise the core just plugs back up.
I went through this the hard way years ago.... on a RAM. The original problem was a heater that didn't put out more than tepid temperatures coupled with intermittent overheat, at least according to the pegged gauge on the dash.
Customer OK'd flush of the heater core, an new t-stat, and a new radiator cap. Problem was only marginally fixed. SO, we put in a mechanical temperature gauge and drove the truck. The temp maxed out at about 240..... right on the cusp of danger to the engine. The true temp at about 225 would peg the gauge, and that is not true overheat on a modern engine, at least not really high enough to do damage.
Problem was, as we dug for info from customer (They rarely tell the entire truth up front) we found out that this was a problem which had existed over several months, and, other shops had attempted repairs (t-stats).
We ultimately figured out that the radiator was restricted, and replaced it. Problem solved..... for about a week... and here it came back with the original problem all over again.... Clogged heater core and recontaminated radiator. HMM!
By this time, the head gaskets were giving up and we were observing froth at the oil fill cap, and slight milkshake color to the oil. SOO, we replaced the head gaskets! During the time said cylinder heads were off, my technician pointed out that the rear coolant passages were totally clogged with cruddy debris consistent with coffee ground like rust scale/ particulate. This clogging was at the rear of the block up to the gasket surfaces of the block... OH! NO!
So, we reassembled the engine, ran the truck over a ditch like depression behind the shop, and I removed the starter and rearmost freeze plugs. I ran a garden hose to the truck, fashioned a nozzle (custom) and crawled underneath and flushed out tons of crud... and got mighty wet and filthy!
Afterwards, we replaced the thermostat with MOPAR part, ditto for the cap, and filled the cooling system. Problem was solved!
As a side note: The truck's owner had been convicted of DWI, and had one of those breathalyzer interlocks that one had to blow AND hum into in order for the truck to start. HOW annoying! I bypassed it for my own convenience, and never dreamed that it would communicate this "bypass" to the powers that be! After a lengthy conference phone call, I got him out of trouble. Those idjits wanted to come after me! I told them that I never had any DWI, or those problems, and refused to let them put me on a guilt trip over it! I did find out that the unit had a 1-800 number on it which I could have called to get a temporary "code" to enter for repair purposes.