How much would it cost to run plumbing 1400 ft to city water? It has to go down a mountain/hill. It might be 800 feet?

Can't say. No one can, without knowing stuff you haven't told us. Best thing to do is have a local contractor come out and look at what has to be done.

Well, if you have a hill, you have to consider the hydrostatic pressure. Figure 0.5 psi for the hydrostatic head, You have not furnished sufficient information, but if the elevation change is 800 feet, you have an engineering problem because you are looking at a 400 psi hydrostatic head. Normal residential pipe will not take that much pressure. Likewise, your 1400 feet on the side of a mountain sounds like rock, so there is no way to furnish you a cost. Now, if you are going to flow volumes for a single residence, say 15 gpm and the elevations are not extreme, a 2" pvc pipe can be 1400 feet and supply water with almost 0 pressure drop. If the ground was almost flat and easy to dig with a trencher, you alone could install the pipe by renting a trenching machine and gluing the pipe joints together. You could do the 1400 feet for about $2 per foot with 2 inch pipe, if you did the labor. It would take a couple of hard days of work and a pickup truck. But the 800 foot hill makes the job beyond anything I know on what to do. Talk to the city and aks them how to do it and the list of contractors that can do this work. It is probably easier to drill a well. You will need electrical power, regardless.