score:6
If you have (or will rent) your own car, it is possible to drive Highway 17.3 between Neum and Hutovo (from which you can proceed to Mostar or other parts of BiH) without entering Croatia. This is bureaucratically the easiest option, so you may want to take it even if you normally prefer not to rent cars.
However, there does not appear to be any public transit along this route, so if you travel by bus, you will indeed have to enter Croatia to reach Neum. If you already have a valid Schengen visa, you can use it to enter Croatia, but if you do not, you will want to apply for a Croatian visa, which is not a Schengen visa and does not grant access to the Schengen area. You will not be able to receive a Schengen visa for an itinerary which does not visit any Schengen states.
Upvote:3
The path you want to take is called "Svitava". I'm not sure if taxi from Sarajevo knows the road but the ones from Herzegovina know for sure. You can try and google it further but here is the road from "Stolac" city.