It's OK as an idea. Cable radars have been in use tens of years.
You can go with much lower spec equipment if you use sinewaves. Connect an easy to adjust rf sinewave generator to the beginning of your test line and put an oscilloscope in parallel with the generator.
You must use frequency so high that your test line surely is longer than half a wavelength. Assume at first propagation velocity 300 m/us. That's the theoretical maximum. Probably the real velocity is much smaller.
Let's assume your test line is 1,5 meters long. Then you are fine with 100 MHz generator.
Have at first a short circuit at the end of the line. If your line has reasonable low losses, you should notice that the voltage at the beginning of the line depends on the frequency. This is because the just produced wave and the reflection have different phase angles. Find the smallest frequency that gives the voltage minimum in the beginnig of the line. On that frequency your line is exactly a half wavelength long. That gives to you the propagation velocity. The result is not accurate until your signal generator is matched to the line ompedance, but that's not your problrm in the question, you wanted the impedance.
If you leave the end open, the same voltage minimum at the beginning should occur at about 50% smaller frequency.
Now try different load resistors (=preferably mass resistors with wires shorter than 2% of your test line, they are not inductive) until you find the one that kills the reflection. That's the line impedance. When you have found it, you will not see any specific frequency that gives a voltage minimum at the beginning of the line.
Instead of trying separate resistors, you also can try a small trimpot that has a flat resistive part.
When you have found the line impedance, you can measure the propagation velocity with sinewaves. Only add the needed extra parallel or serial resistor to make the beginning of the line matched and find the frequency where your line is half a wavelength long.
NOTE: You must use a high z probe in the oscilloscope to prevent it to disturb the measurement. The cable that connect your signal generator to the test line must be only few centimeters long if you operate at 100 Mhz maximum.