Well with a cursory glance perhaps I can help if its not too late but is there some missing information?

It sounds like your X is composed of some discrete values.

The expected value I can compute now (based on wikipedia for 'expected value'):
its just the integral of the x*f(x) = -ln(x) + x - 1/x

The last two parts of the question are easy, since you have the PDF you know the probability of any x occuring. Take and solve for 3. Then solve for the boundary conditions 1 to 3 with the CDF.

The integral is:

solving for x=3 yields a probability of 26% or so.
using ln(x) + 1/x - 1/(2x^2), and ntegrating from 1->3 yields a prob of 88% or so.

I'm not sure on the mode or median (I feel like this is where information might be missing) but the mean is the point at which the CDF = .5 (when x=1 according to my calculations). I don't know if that helps with the problem, but an fyi.