Someone mentioned brake cleaner - I think that brake cleaner and brake fluid are both too corrosive to use as a wash or cleaner for a manifold. I know that there are those who are proponents of using brake cleaner for cleaning out your heads, but I am not one of them.
The original question, or one of them, asked how much baking soda to use to neutralize vinegar. The answer is "as much as it takes to neutralize the acid" I am not being a smart aleck, the acid strength has been changed by its use, it wasn't that precise out if the bottle, and nobody can give you an exact amount of baking soda to use. Best direction is to add until no more fizzing. Or measure until ph = 0. Vinegar isn't a problem to neutralize anyway - it is so weak you can drink it. Let the gunk settle down to the bottom, pour the clean stuff off the top and save it. Neutralize the remaining gunky stuff. Most of the vinegar was water, let it evaporate if you have a lot of it. Then get rid of it with the rest of your oil waste. The only bad stuff is the part you crudded up with your iron and carbon.
I think you need to scrape where you can and not worry about light carbon buildup in the head. If you have so much buildup inside the head that it is a problem you need to have it professionally cleaned. If it is light buildup, change to a high quality synthetic oil. Or add some treatment to your gasoline to clean it up.
Get it as clean as you can by scraping, wire brushing where you can, etc. Then put it back on. If you are real concerned that it is choking off one of the channels, wire up a little rag and pull it through there.
If you want to clean it just because you want everything factory clean, have it done by pros.
There us no "cheap" way. I would almost bet money that any performance issues are more related to quality of fuel or air/fuel ratio than any buildup inside head tunnels.