Might be more than one way to approach this, but do note that there are some existing map scripts/mods out there that have civs in their starting locations. For the delivered map, all I can suggest right now is to create a scenario from the map. Start a custom game with the map > select the civs you want > run game > open WB > I think you should then be able to plant a starting city for each civ where you want them. Delete the start units, of course, as they will be in random positions. Then select a civ > plant a city (i think it is the first option on the terrain tab but forget) on the tile you want them to start on > add starting units > select next civ > repeat. Then save the scenario file in WB. I believe that scenario should then play each civ in the starting location you planted the city. Not sure, but it might work by just moving the starting units to the appropriate spots without planting a city, but for your purpose planting a cap city in the WB file should work fine. If that does not work, I suggest posting the question in the Mod forum (C&C) cause someone is sure to know there. Plenty of mods out there have scenarios with static start positions.
edit: If you are starting with an Earth "scenario" like Earth18, you cannot change the civs from in-game. You can from notepad. I suggest saving off the scenario from WB so as not to overwrite the original. WB scenario files can be opened in Notepad and edited. In notepad note the Team #s and match them with the corresponding settings, i.e, the first list has things like starting techs and civs met (usually themselves), and the next has much more civ-related stuff. You can overwrite the text here to replace one civ with another. If unsure of the code for each civ, try running a separate game with the civs you want > save scenario in WB > open that in notepad to get the proper code. Make sure you change all the necessary fields for the civ and make sure they have the right starting techs above that for the same team. Here is an example:
It should be pretty clear here what fields to change by comparison to the dummy scenario. Note that each team section here has set start positions by x and y that fixes them to the spot. That is one way to do it but takes a bit more work, but you can get the x/y position for a tile by hovering over it I believe(in-game). If say, you want to change just the leader to Ramses, just change the LeaderType and LeaderName. Or you can replace the civ entirely but change the obvious fields based on a comparison the what the civ you want looks like in the dummy game.
You can also delete a team/civ entirely, but just make sure you adjust the numbers and remove the team entirely in both spots. Or just change Team 0 - X and delete the rest. Or add more if you wish.
Lastly, note that the starting units are set further down in the file and correspond to the starting x/y position of the civ. You can find them by searching on "unitowner" and just matching it to the team.e.g., unitowner=1 is Team 1's starting units. Adjust, reposition or delete as needed. Not sure how that is affected by difficulty though, so play around with it to see if it will still adjust with difficulty - not sure how that works exactly.