Low-dimensional carbon allotropes have attracted much attention due to their great potential for applications in future electronic and magnetic devices. Recently, new graphene-like nanoribbons consisting of four-, six-, and eight-membered carbon rings have been synthesized successfully, and a related two-dimensional structure net-Y was subsequently proposed. Here, we report theoretical investigations on the structural stability and magneto-electronic properties for the net-Y armchair nanoribbons doped with low-concentration transition metal (TM) atoms (Mn, Ni, Fe, Co, and V), especially focusing on their electric-magnetic and mechano-magnetic coupling effects. The calculated binding energy and molecular dynamics simulation confirm that all these ribbons hold a high stability. Particularly, by TM-doping, the intrinsic non-magnetic semiconducting ribbon can be converted to a bipolar magnetic semiconductor (BMS), a half metal (HM), or a spin-degenerate magnetic semiconductor (MSC), presenting rich magnetic features. Meanwhile, this magnetism is flexibly tunable by the transverse electric field or stretched strain. For example, a low electric field can transform the ribbon into the HM phase, and the mechanic strain can realize a continuous magnetic phase transition among BMS, magnetic metal (MS), MSC, HM, and half semiconductor. These findings indicate that the TM-doped net-Y ribbon possesses favorable magnetism, which might have promising applications in magnetic nanodevices.