Have not come across a Hadith which mentions a specific manner that Nabi (sallallahu ‘alayhi wa sallam) applied ‘itr. However, based on the following Hadith, one should start with the right hand:

Sayyidah ‘Aaishah (radiyallahu ‘anha) said, “Nabi (sallallahu ‘alayhi wa sallam) loved to start activities from the right side whether it was wearing shoes, combing his hair, or performing wudu and all of his matters”(Sahih Bukhari, Hadith: 168)

 Nabi (sallallahu ‘alayhi wa sallam) would apply ‘itr on his head and beard.(Refer: Muntahas Sul, vol. 1 pg. 356)

 And Allah Ta’ala Knows best.

3 links below

https://hadithanswers.com/sunnah-method-of-applying-itr-fragrance/
https://askimam.org/public/question_detail/22506
https://darulifta-deoband.com/home/en/qa/12674