A glass holds 0.35 kg of water at 28 °C…

Step 1. Can it melt? Melting needs $0.060\ \text{kg} \times 3.34 \times 10^{5}\ \text{J/kg} = 20\,040\ \text{J}$; the water can give $0.35\ \text{kg} \times 4186\ \text{J kg}^{-1}\text{K}^{-1} \times 28\ \text{K} = 41\,020\ \text{J}$ before reaching $0\ ^\circ\text{C}$. So yes.

Step 2. Balance. $41\,020 - 1465T = 20\,040 + 251T$, so $T = 12.2\ ^\circ\text{C}$.