How do I find the slope of the tangent line at a point, given a graph without being given the numbers of the point or the equation of f(x)?

Without point or function information, the best you could do is an estimate. Hopefully this graph would at least have a numbered scale on the coordinate axes. You would draw in a tangent line at the point you are interested in, then estimate the change in y (rise) and the change in x (run) to get the slope estimate = (rise / run).

For example, a portion of your drawn-in tangent line might rise 3 units over an x-interval of 2 units, so the slope estimate would be 3/2.