Database
[{arrangements:[{displayName:"MRT.8"},{displayName:"MRT.10"},{displayName:"MRT.12"},/**(...)*/]}]
Query
db.collection.update({"arrangements.displayName":{$regex:"^MRT."}},[{$set:{"arrangements":{$map:{input:"$arrangements",in:{$mergeObjects:["$$this",{displayName:{$concat:["MOBILE.",{$arrayElemAt:[{$split:["$$this.displayName","MRT."]},1]}]}}]}}}}}])