Latitude is relatively simple. I'll just follow the sun's pattern and, with the help of increased time here, I can determine the shift of the axis depending on how the seasons and night cycles change, granted if we're on this planet's equivalent of an equator, it won't shift too much if at all (depending on the angle).
So... yes, I can figure it out, using the astronomy methods of ancient times. Using a telescope would be more beneficial because then I could triangulate the distance of us from the sun, then use how fast the sun moves across the sky, to formulate just how fast the planet is spinning. Then I could even do longitude.