This　paper　focuses　on　the　initial-　and　boundary-value　problem　for　the　two-dimensional　micropolar　equations　with　only　angular　velocity　dissipation　in　a　smooth　bounded　domain.　The　aim　here　is　to　establish　the　global　existence　and　uniqueness　of　solutions　by　imposing　natural　boundary　conditions　and　minimal　regularity　assumptions　on　the　initial　data.　Besides,　the　global　solution　is　shown　to　possess　higher　regularity　when　the　initial　datum　is　more　regular.　To　obtain　these　results,　we　overcome　two　main　difficulties:　one　due　to　the　lack　of　full　dissipation　and　one　due　to　the　boundary　conditions.　In　addition　to　the　global　regularity　problem,　we　also　examine　the　large　time　behavior　of　solutions　and　obtain　explicit　decay　rates.