Does Doppler shift affect apparent pulsar frequency?
Yes, it is correct.

Consider the classic example of a (say) $1000$ Hz siren on an ambulance approaching you.  Once a single wave is produced, the source moves towards you before the next wave is produced.  The wavelength of the sound waves in the air is reduced, so the frequency you hear is increased.  Simple, basic Doppler shift...

But consider the case where the siren is being switched on and off with a five-second overall period.  So you have a $1$ kilohertz signal modulated at $0.2$ Hertz. 

But the same argument applies to this modulation frequency.  A chain of $1$ kilohertz waves is generated for $2.5$ seconds and sent on its way.  The ambulance approaches you for the next silent $2.5$ seconds, then starts the next chain on its way, closer to the last chain because of the distance travelled in the silent phase.  The same Doppler shift factor applies to the carrier frequency and the modulation frequency.

The same physics almost derailed the Cassini-Huygens probe. The extreme velocity changes of the probe caused a Doppler shift in both the transmission frequency and the data rate carried by this transmission. The first was handled;  the second was not!

See http://www.thespacereview.com/article/306/1

or

http://spectrum.ieee.org/aerospace/space-flight/titan-calling