How to show special respect to a famous PhD advisor?
I don't know what field you're in, but in mine, writing such a paper would be viewed as unnecessary and perhaps a bit strange. You seem to feel that you don't deserve your place with this supervisor, and so you need to prove your worth, or impress them, by writing a paper. This is unecessary, as you have already impressed them with your project idea and PhD application. They would not have taken you on as a student otherwise. Along these lines, you might want to search the term &quot;impostor syndrome&quot;, both on this site and on Google.
&quot;I am exhausted...&quot;. The last thing you need to be doing right now is writing an unnecessary paper with no help from anyone. A PhD is a several-year-long marathon, and you will need all your energy and stamina to complete it. Take a break, rest and recharge before you start. This will have a far greater benefit to your PhD, and accordingly to your relationship with your supervisor, than writing a paper.
Once you have started the PhD, show your current paper draft to your supervisor and work on it together. The paper will be much better for it. Again, this is field dependent, but in mine it would be extremely rare for a beginning PhD student to be capable of writing a publishable paper entirely on their own. You may be underestimating the challenge. This is something that your supervisor will help you with.
But for now, have a holiday, and celebrate your success at being accepted for a PhD with a great supervisor!