How can I detect a pulse from a device with the AC'97 component of a Xilinx Atlys board?
Without more information all we can do is guess. You don't come right out and say whether this source of "rapid pulses" is AC'97 so my answer is going to assume it's just some arbitrary pulse train you want to look at and process. If it's AC'97 then you probably don't want to connect it to a (presumably) analog audio input.

Guess 1

The 3.5mm jack on the Atlys board has some minor signal conditioning and can be connected directly to an input pin of the FPGA. You can run that signal through a synchronizer and Bob's your uncle. Your detection time will have a resolution of the FPGA clock signal used to synchronize the incoming signal, and will have delayed by however many flip flops are in your synchronizer.

Guess 2

The 3.5mm jack on the Atlys board is treated as an analog signal (makes sense, as 3.5mm jacks are typically used for audio), conditioned as such and fed to the input of an ADC.  In this case you've got to drive the ADC with the FPGA and look at the output of the ADC to determine when a pulse came in. The signal's bandwidth is only 6kHz so this really shouldn't be an issue, and you can filter the ADC data more if need be. The output of all of this should be a digital pulse train.

Guess 3

Similar to Guess 2, but maybe you're lucky and you have the analog signal from the 3.5mm jack feeding a comparator. The output of the comparator goes to the FPGA, and now you've got a digital signal that must be synchronized and used just like Guess 1.

Once you have this digital pulse train in the FPGA you probably want to count pulses over some period of time. If it's an AC'97 stream there are plenty of resources on decoding AC'97, but let's ignore the content of the data stream for the moment.

If you're just counting pulses something like this might work:

if pulse_bit changes
    if pulse_bit = '1'
        ++pulse_count
    end if

    reset_pulse_timeout
end if

if pulse_timeout expires
    copy pulse count to toplevel
    reset_pulse_timeout
end if


If you're actually looking to extract some kind of intelligence from the pulse train (i.e. it's not just a pulsing line but more like a serial line and you're looking to get data from this) then you want to look at various async or sync serial ports (UART, TDM or USART) to look for the start of a transmission, grab bits and look for a stop bit. At that point you now have a data word you can look at specific bits within for what you want.