No, it’s a Japanese Garden not a park per se so nobody jogs there. Try Yoyogi Park or the jogging route around the Imperial Palace.

As for "pre booking" as I say, as a public park operated by Tokyo Metropolitan Government - as such, and as a public facility, there is no "booking" or "pre booking".

During peak seasons: 1. first week of April for sakura cherry blossoms and 2 late November early December for red maples there will be extended night viewing. For some reason this is popular, but the glare from the lighting and the mob scenes are a real turn off, but for some reason Japanese get off on artificially illuminated cherry blossoms and maple trees and don't mind being jostled by crowd scenes. Count me out. Tokyo Metropolitan Parks manage these gardens, just google them, easy to find. They might not update with extended hours for sakura or maples, but phone numbers are given and your hotel concierge or a Japanese friend can ask about night viewing time.