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The short answer is yes.
Whether you will pay anything other than the standard undiscounted fare is debatable.

I have to ask why you would want to do this.
Is it because you are not sure exactly when you might arrive at Dover having travelled across the Irish Sea and then across England?

You could book a ferry and then just turn up late or early as long as it is the same tariff sailing.
We had a problem on one occasion and couldn't get away from home on the correct day. We just turned up exactly 24 hours late, explained we'd had a last minute hitch and they put us on the next ferry.
What they don't accept is if you turn up for a premium rate ferry having booked a cheap/off-peak. They think you are trying it on (which many do) and make you pay the extra to go.

 

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We've often been sent on an earlier boat when we arrive to book in.
We've often thought that if they have the space and send us early they are creating a space on the next ferry which they can sell to anyone just turning up.

Similarly from Calais.
We arrived at the port 2 days early when we came back from France on 3rd May but they put us on. Again we turned up for the same time ferry although the wrong day.

 

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Just found this: http://www.dovereurochange.com/ with map.

I see that they are open 08.00 to 18.00

What a pity they aren't open an hour or two later.
We could have gone to our overnight spot on the Marine Parade opposite and walked over and booked for next morning.

Still well worth knowing.
Thank you Jon.