According to Dvorzsák's gazetteer (1877), Maros-Újvár in Alsó-Fehér county had a Greek Catholic, Reformed, and Jewish congregation locally, while Roman Catholics were recorded in Felvincz, Lutherans in Nagyenyed, and Unitarians in Szentbenedek. The civil registration office was local. (Civil registration began Oct. 1, 1895; before that, vital records were kept by churches.)
As you note, the place is now Ocna Mureş, Romania. This unfortunately means that there is little or nothing available online for documenting genealogy from this place. FamilySearch has no catalog entry for Marosujvár (under any name), and it was a primarily Hungarian town, not a Transylvanian Saxon (German) city, so Ancestry is unlikely to have any records for it. (I don't have a subscription to check.)
A few websites to check:
Romanian National Archives: http://arhivelenationale.ro/site/en/first-page/
Transylvanian Family History Database: http://www.familyhistory.ro/index.php?id=20061203transylvanian_fa
Hungarian Family History Society's list of ecclesiastic archives in Romania: http://www.macse.hu/society/romania.php?id=egyh