Monolayer (ML) Ga2O3 with wide bandgap and ultra-high electron mobility has gained extensive interests due to its great potential in next-generation electronic and solar-blind optoelectronic applications. Here, we perform a comprehensive investigation on the electronic properties of the ML Ga2O3 field-effect transistors (FETs) with different metal electrodes spanning a wide work function using energy band structure and quantum transport (QT) calculations. The results indicate all the investigated metals form n-type Ohmic contact with ML Ga2O3 in vertical direction, while high Schottky barrier heights (SBHs) exist for holes except for Sc due to its metallization and re-construction at the contact interface. In addition, the existence of tunneling barrier (TB) at the interface of ML Ga2O3/Ag, Au, Pd, and Pt systems reduce the electron injection efficiency to 20.09%, 13.03%, 34.70%, and 26.29%, respectively. Using a back-gated FET configuration, Al electrode exhibits the highest performance with Ohmic contact property and absence of tunneling barrier, thus is believed to act as the best candidate electrode material for ML Ga2O3 transistors. In addition, the results indicate that the current of ML Ga2O3 transistor is dominated by electrons rather than holes. Our study offers a theoretical foundation for the electrode selection for ML Ga2O3 devices.