Rukavytsya scored the winning goal in the 71st minute, after a pass by the substitute midfielder Dolev Haziza.
The 29-year-old, who helped Bayern to the treble last season, arrived in Liverpool on Friday morning to complete his medical and put pen to paper.
Al-Taawoun lost its last match to Iran's Persepolis 1-0 on Tuesday, but is still atop of Group C with six points from three matches.
The country is undergoing an uncertain weather condition, which may render the fields to be unsafe.
Athletico capitalized on the numerical advantage when Walter, unmarked inside the 18-yard box, struck his first goal of the campaign.
In men's 5,000m, Duo Bujie became front runner from halfway, but he was overtaken with two laps to go. On the final lap, Peng Jianhua emerged to win with 14:13.78, Qin Zhenfei and Zhao Changhong finished second and third respectively, Duo Bujie missed the podium with the fourth place.
The 22-year-old Ge set a new personal best of 22.69s this May, aiming to qualify for the postponed Tokyo Olympics final. "I was too hasty to concentrate, when I heard some noise I thought it was the gun," she said.
In the gala, Shaanxi announced five image ambassadors of the major tournaments. Olympic diving champion Qin Kai is in the list. Besides him, Chinese famous actress Yan Ni, 'anti-pandemic hero' Han Jianfeng, Tai Chi player Wu Yanan and swimming coach Liu Nan are in the roster.
It added that a meeting would be held with FIFA officials on Tuesday to address issues related to the competition "at the highest level."
The game's turning point came in the 86th minute as Bi Jinhao tapped Yu Hanchao's cross to level the game for Shenhua, and that was just the right reward for Shenhua head coach Choi Kang-Hee, who made a raft of changes midway through the second half. Enditem
"Staying at the top might be much harder than getting to the top. One has to consider all the possible details both on and off the pitch."