How to estimate the analog bandwidth?

If your scope's input amplifier has a frequency response of a first-order RC-filter, you can roughly estimate the bandwidth from the rise time:

$$BW ≈ 0.35 / t_R$$

To clarify, bandwidth is defined by the frequency which is attenuated by -3dB, and the rise time corresponds to the input signal going from 10% to 90% of its amplitude.

Of course, this only applies when you're sure that the observed rise time is due to your scope delaying the signal which originally is (close to) an ideal square wave. If your input signal has a known rise time itself, it should be subtracted from the measured rise time before applying the formula. At 500kHz however, I expect your square wave to be very close to ideal, compared to the rise time you observe with your scope.

It's very hard to say whether your estimation is right without knowing more about the system and the input signal. Looking at the rise and fall times it seems reasonable by eye, but if you want a good estimation of bandwidth, it makes much more sense to use a sinusoid waveform rather than a square one. With the square input your effectively checking the slew rate, but you can't be sure how much of the slew rate limiting is happening because of your source and how much is happening because of the scope.

By sweeping the frequency of a sine input, you should be able to monitor the frequency at which the displayed amplitude drops by 3dB (voltage amplitude becomes \$1/\sqrt{2}\$ ), which will give your -3dB corner frequency. You will also be able to measure the rolloff if you're so inclined, this will depend on how you have designed the input stage of your scope.

You say you have no specialized equipment, so assuming you don't have a sinusoid function generator, I would suggest building something like a Wien Bridge Oscillator. This will give you a neat sinusoid source, with only a few components. By changing the resistor values, you can get different frequencies for your sweep. If you don't have a small low voltage bulb, there are other designs which don't need it (you lose a bit of sine linearity though).