Can I suggest Stellarium, which is a free planetarium program available for Mac OS, Windows and linux.
For any date, time and geographical location, you can obtain sunrise time and position.
To take account of mountains etc, there is the added option of including into Stellarium a horizon image. This will give you an idea of how mountains etc may affect sunrise time.
I rather think, however, that including a horizon image into stellarium might well be more effort than you intend to exert!
As a rough and ready estimation of the angle subtended by a mountain, estimate as follows from here:
You can estimate altitudes by extending your right arm in front of you, at eye level, with the hand bent toward you, and turned so that the palm and fingers are extended horizontally to your left. With the thumb held next to the fingers, so that all five digits are close together, the width of the hand is about 10 degrees. If you have a narrow hand, or a long arm, your hand may only cover 9 degrees, and a wide hand, or a short arm, may cause your hand to cover 11 degrees; but there isn't generally much more variation than this.
You can then use the sun altitude data from stellarium to estimate when the sun will appear over the mountain. You might also need a compass to orient yourself and your observations.