It's easy to lose the drive in a job you know you are leaving, but it's important to be professional and keep going until the end (especially if you would like the end to come an additional week earlier!). If you still have annual leave left to use up, it may be better to take them for long weekends or the occasional day off here and there, rather than a single week-long chunk. It may help with the motivation; to work only a few four-day weeks during the notice period.
I would be surprised if I was tasked with recruiting my replacement, but it gives you an opportunity. If New Guy can start during your notice period, that may help your case of taking a holiday in between jobs. Is there even the option of taking the time off unpaid?
If you have negotiated a whole month off your notice period, you probably would be pushing your luck to shorten it further, especially for a small start-up company! If your replacement starts while you are still there however, you could maybe speak to your boss about allowing you and New Guy to spend time on some knowledge transfer. It will allow you to wind down on your other responsibilities while confidently preparing New Guy to continue without you. This would also instil some confidence in your boss that they could do without you a week early, assuming New Guy rises to the role.