Leave aside the cost of barrel installation - that's the same whether you are installing the barrel on a Remington or on a custom action.
There's probably no such thing as an "average cost" to blueprint/true up a Remington. Different smiths will do different things, will take more or less time to do them, and charge more or less per hour. You could easily see prices in the $100 - $500 range.
In most cases, my opinion is that you are better off to buy a better action, rather than improving an existing one. Don't get me wrong; properly done, a blueprinted Remington (or other) action can shoot fully as well as a custom action. But in the end, it's "just" a trued-up Remington. If you ever try to sell it, its market value will likely be a lot less than the sum of your purchase price and the work needed to get it fixed up.
Custom actions aren't terribly expensive these days; we are quite spoiled, we have a great many very good actions to choose from.
I would suggest that you figure out what you want (single shot, or repeater? Better-than-bone-stock accuracy, or topnotch match-winning accuracy? etc). Then figure out the best way to achieve that.
If you haven't already tried shooting a variety of long range rifles, it would be a good idea to try various ones ASAP - that will help you better decide what it is you really want.