What is the wing angle of attack of a Boeing 737 in cruise?
First, let's assume that all lift is created by the wing, and that the plane is flying in ideal conditions, where fuel flow per distance flown is minimal. This point is given by $$c_L = \sqrt{\frac{1-n_v}{3+n_v}\cdot c_{D0} \cdot \pi \cdot AR \cdot \epsilon}$$ Please see this post for more background on this formula.

Nomenclature:
    $c_L \:\:\:$ lift coefficient
    $n_v \:\:\:$  thrust exponent, as in $T \sim v^{n_v} $
    $\pi \:\:\:\:\:$  3.14159$\dots$
    $AR \:\:$  aspect ratio of the wing
    $\epsilon \:\:\:\:\:$  the wing's Oswald factor
    $c_{D0} \:$  zero-lift drag coefficient

The $n_v$ I would estimate to be -0.4, the $c_{D0}$ is somewhere around 0.016, the aspect ratio can be taken from this source and the Oswald factor is 0.8. This gives you an optimum lift coefficient of 0.445. Now we need to know what angle of attack is needed for this lift coefficient. If we look at the midspan airfoil of the 737, this angle of attack is around 2.8 degree.

This source is for model airplanes, so their Mach and Reynolds numbers are much too low for the real thing, and we need corrections for both. Using the Prantdl-Glauert rule for Mach effects, we need to correct by dividing with $\sqrt{(1-Ma^2) \cdot (1 + tan\varphi)}$ = 0.693 at Mach 0.82. Please note that this increment must be applied to the zero-lift angle of attack, which is -1°. Now our angle of attack is only 1.63°. The flight Reynolds number of the 737's MAC is 22 million, so the angle of attack is even less, maybe 1.5°. Now we need a correction for the 737 wing sweep of 25 degree, which is computed by dividing the angle of attack by 0.9, the cosine of the quarter chord sweep angle. This gives about 1.8° as the corrected value.

So my quick-and-dirty answer is 1.8 degrees. Other sections of the wing have slightly different values, according to the wing's twist. Please note that the wing area of the 737 is roughly constant since the Sixties, while the mass has increased considerably. It is safe to assume that it is operating at higher angles of attack than the optimum for best range.