Would it be worth it for Sunspear to enhanced its harbor and use the Greenblood to pull trade away from Oldtown? What would a ruler have to do and how soon could such a project be pulled off?

I don’t think so. 

First, Sunspear isn’t on the Greenblood (it’s a good bit north of the mouth of the Greenblood), so it would really make more sense to build up Plankytown (which sits at the mouth of the Greenblood. 

Second, as far as I can tell, there is no harbor at Sunspear. Indeed, when “Princess Nymeria and her ten thousand ships made landfall” near Sunspear, she did so “on the coast of Dorne,” beaching her ships rather than anchoring them at a harbor.

Third, it’s not clear how easy it would be to pull trade off of Oldtown from such a location. Yes, you have the Greenblood, but those waters reach their terminus in the middle of the Dornish peninsula and it’s a long caravan route indeed to get into the Reach or the Stormlands. By contrast, Oldtown is located in a much more populous market and has access to the great heart of the continent, either taking the Roseroad up to Highgarden (which then gives you access to the Coast Road and Casterly Rock) and then to King’s Landing, or taking the Mander as far north as Silverhill and as far east as Tumbleton or Grassy Vale.