Was there a route via the north of Russia that Admiral Rozhestvensky could have passed (maybe during the summer months) instead of the seven month odyssey of going through Cape of Good Hope around Africa and then through Indian Ocean and South China Sea to reach Vladivostok and eventually Port Arthur.
Absolutely no. The first time when the North-East passage was made in one navigation (that is in less that one year) was in 1932 using icebreakers. And this was just an experimental expedition. In 1905 Russia had no enough icebreakers of sufficient power, not even mentioning other difficulties, like coaling stations on the way.