Alias for root of a polynomial

You can give u an UpValues for Power:

u /: u^n_Integer := Block[{u},
    If[n<0,
        PolynomialMod[(-u-1)^-n, 1+u+u^2],
        PolynomialMod[u^n,1+u+u^2]
    ]
]

Then:

y = Series[u + 1 + u x + x^2, {x, 0, 4}];
z = Series[u^2 + u^2 x + x^4,{x, 0, 4}];

and:

y + z //TeXForm

$-x+x^2+x^4+O\left(x^5\right)$

You can use Assumptions

assume = u^2 + u + 1 == 0;

y = Series[u + 1 + u*x + x^2, {x, 0, 4}];
z = Series[u^2 + u^2*x + x^4, {x, 0, 4}];

Assuming[assume, SeriesCoefficient[y + z, 0] // Simplify]

(* 0 *)

Assuming[assume, SeriesCoefficient[y + z, 1] // Simplify]

(* -1 *)

The simplest methods are usually the best. I suggest

rule = {u^n_ :> {1, u, -1 - u}[[Mod[n, 3] + 1]]};
y + z /. rule

which will do what you want. Also, the following code

Table[u^n, {n, 0, 6}] /. rule

demonstrates that $u^3 = 1$ and the powers of $u$ are periodic with period $3$.