Assuming that the probe is an MAS or solenoid coil type, you can make use of the geometry of the sample. Both of these have typical sample volumes that are longer in one direction. You can use a 1-D projection imaging technique to find the coil axes.

Examine the probe and make a mark on the bottom of the probe (where it will still be visible when the probe is in the magnet) indicating the long axis of the sample coil. Fill the sample volume with water (or paraffin for protons, or something else that gives a good SNR on a single pulse). Put the probe in the magnet and establish a pulse sequence (e.g. 1pulse) which gives a good signal. Ramp the X (or Y) shim to a large value. Record the spectrum again. Note the linewidth. Now rotate the probe in small increments, taking a spectrum after each rotation. The minimum linewidth will occur when the sample is perpendicular to the X (or Y) axis. You can then mark the probe mount, by reference to the mark you made earlier on the probe.

For best accuracy, you should probably plot the linewidths vs. rotation angle. Then you can smooth and interpolate the curve. You could get a plastic protractor from the bookstore, cut out the center, and fasten it to the bottom of the magnet as a reference for the angle measurements.