How to solve for voltage using node voltage method?
I don't know what your instructor considers the "node voltage method", but this is easy to solve.

First note that turning on each voltage source results in some positive step on Vx, and that each positive step can be computed independently and then summed to get the net step on Vx when the supplies are turned on.

Next note that for the purpose of analysing the result of a single supply stepping on, the other three capacitors are in parallel connected to ground since all the supplies have 0 impedance (by definition of a voltage source).  So for each supply, the step on Vx is the result of a capacitive voltage divider between the capacitor in series with that supply and the parallel combination of the other capacitors.  The absolute value of the capacitors don't matter, just their ratios.

The 5V step is divided by 50aF and 120aF.  5V * 50 / 170 = 1.471V.

Left 1V step: 1V * 10 / 170 = 59mv

Bottom 1V step: 1V * 100 / 170 = 588mV

2V step: 2V * 10 / 170 = 118mV

Adding up all the contributions yields a step of 2.235V.  Adding this to the initial state of 2V yields the final answer of 4.235V.