Since you're talking about a quantum mechanics book rather than a radar design engineering book, I assume that they're talking about a basic textbook radar system, rather than one with all the advanced features.
The receiver needs to have a bandwidth great enough to receive most of the energy in the reflected pulses. Since they're sine waves modulated by a pulse shape, they will have spectral splatter and the receiver needs to be able to process most of this energy for each pulse, which is now spread over a range of frequencies. You're right in that, to compute this spectral spread, you need to do a Fourier transform - the spread you'll get depends on the pulse shape.
The longer the pulse, the narrower the bandwidth of the spectrum - sometimes people use a "rule of thumb" whereby the bandwidth is estimated as the reciprocal of the pulse duration.
With respect to range, I suppose if you imagine two targets close to each other(let's assume same azimuth), one behind the other, then you can resolve them if the reflected pulse from the nearest has finished being received before the reflected pulse from the furthest starts to be received. It's fairly easy to see that this implies that the distance between the targets must be greater than 0.5*PulseTime*c