A uniform answer for all such cases is going to be too much to expect when $n > 2$.

To begin with, $w$ may not even *have* a continued fraction expansion - it may fail to be real! (But at least the conjugates of a real algebraic number will also be real under the given conditions.)

And then the relationship between $w$ and any one conjugate may be arbitrarily complicated.

More can be said only when you restrict your attention to particular families, such as for example Shanks' "simplest cubic fields" defined by polynomials $X^3-TX^2-(T+3)X-1$ for integer $T \ge -1$ where the automorphisms can be expressed by a single formula independent of $T$.

As to your second question: Nowadays given a defining polynomial $f$ for (real algebraic) $w$, you would use your favorite computer algebra or computational algebraic number theory software package to compute a sufficiently accurate approximation, and then turn this into the desired number of partial denominators.

However, there is also a direct algorithm starting from the integer coefficients of $f$ and working only with (large) exact integers: Begin by bracketing the real roots of $f$ between successive integers using Sturm sequences. Then when you know that, for some $a\in\mathbb{Z}$ and real $\delta > 1$, one root can be written as $f(a+1/\delta)=0$, you can rewrite this as a new polynomial equation $f_1(\delta)=0$ with integer coefficients depending on $a$ and on those of $f$. Repeat as often as you like. Serge Lang and Hale Trotter were using this approach long before computer algebra software became widespread ("Continued fractions for some algebraic numbers," J. f. r. angew. Math. **255** (1972), 112-134; Addendum: ibid., 219-220). I'm not sure whether that article is available online, but you'll find a very readable introduction by E.Bombieri and A.van der Poorten here.

Edit: A scan of the Lang-Trotter paper is available online, too. They do discuss the case $T=-1$ of Shanks' family in the guise of $2\text{cos}(2\pi/7)$.