Big O Notation of $n^{0.999999}\log(n)$
If $g = O(g')$ and $h$ is nonnegative then $h \cdot g = O(h \cdot g')$ — in English, if $g$ grows slower than $g'$ then multiplying by a positive function doesn't affect that fact. (If you aren't sure about this, do the proof with quantifiers).

Since $\log(n) = O(n^{0.000001})$, $n^{0.999999} \cdot \log(n) = O(n^{0.999999} \cdot n^{0.000001})$. Since $n^{0.999999} \cdot n^{0.000001} = n$, we have $f_1(n) = O(n)$.

While it is true that $f_2(n) = O(n)$, this is not relevant here. What you need is $n = O(f_2(n))$, which is also true.

Intuitively, a higher exponent trumps a logarithm. A multiplicative constant is not significant.