I am thinking to do first making mono acid than acid chloride and finally amide from acid chloride.
There is quite a lot of literature precedent for the ester hydrolysis - the ester
trans to the vinyl group is selectively saponified in high yield with NaOH (
Example [see supplimentary info]).
To play on the safe side, you could then do a DCC etc.-mediated amide coupling or activate the acid as a mixed anhydride with isobutyl chloroformate (the conditions for these reactions are milder than those used to make acid chlorides).
You could attempt selective amidation of the less hindered ester in one step by heating the diester with the amine directly - hydroxide attacks the ester
trans to the vinyl group with high selectivity, so it is probably a reasonable assumption that an amine nucleophile might behave similarly. I would test this direct approach first because it is a much shorter route (but the feasibility depends on the bulk of the ester and the amine, which you have not mentioned).