Hi there,
Interesting question! Let's have a look at normal JR ticket prices and find out.
Osaka - Arashiyama (return) ¥ 1,940
Osaka - Kinosaki ¥ 5,800
Kinosaki - Tokyo ¥ 17,290
Tokyo - Nikko ¥ 5,780
Nikko - Tokyo ¥ 5,780
For a total of ¥ 36,590 or ¥ 7,490 in savings, as the 7 day JR Pass is ¥ 29,100.
Note that between Tokyo - Nikko there are also lower cost routes available, however with the JR Pass you can use the Shinkansen and travel much faster to Nikko. In addition the JR Pass may also be useful for travel from/to the airport and for travel within Osaka/Tokyo adding even more to the above savings. Also remember that with the JR Pass, you'll only need one ticket for all your travel.
Hope this helps!
Daniel