Simple Voltage Calculation
According to Kirchhoff's Current Law, at a point, the sum of incoming or sum of outgoing currents must be zero (unless there is a charge accumulating on that point, and this is not our case).

We take the red point in the bottom as our reference point; we will assume that its potential is zero volts. This is just OK since we are only concerned with the potential differences rather than the actual potentials; and also your question is asking the voltage across a resistor, not potentials at its terminals.
Next, we define another node (X - the top red point), and call its potential Vx. Our aim will be forming an equation with this unknown Vx variable, and then solve for it.

So, we write the sum of currents at the X point:

$$ I_1 + I_2 + I_3 = 0 \dots (1) \\
I_1 = \frac{V_x - 5V}{150 \Omega} \dots (2) \\
I_2 = \frac{V_x}{100 \Omega} \dots (3) \\
I_3 = \frac{V_x - 0.45V}{12 \Omega} \dots (4) $$

Substitute (2), (3) and (4) into (1):

$$
\frac{V_x - 5V}{150 \Omega} + \frac{V_x}{100 \Omega} + \frac{V_x - 0.45V}{12 \Omega} = 0 \dots (5)$$

Rearrange the terms in (5):

$$
30V_x - 21.25V = 0 \\
V_x = 0.7083333...V $$

You were right. Since Vx corresponds to the voltage on the 100\$ \Omega \$ resistor, it indeed equals to 0.708V.