To solve the binary linear equation $2x+y=6$ and find the correct solution among the given options, we substitute each option into the equation and check for equality.

Option A: $\left\{\begin{array}{l}{x=2}\\{y=2}\end{array}\right.$

Substituting $x=2$ and $y=2$ into the equation, we get:
$$2 \times 2 + 2 = 4 + 2 = 6$$
Since the left side of the equation equals the right side ($6=6$), option A satisfies the condition.

Option B: $\left\{\begin{array}{l}{x=1}\\{y=3}\end{array}\right.$

Substituting $x=1$ and $y=3$ into the equation, we get:
$$2 \times 1 + 3 = 2 + 3 = 5$$
Since the left side of the equation ($5$) does not equal the right side ($6$), option B does not satisfy the condition.

Option C: $\left\{\begin{array}{l}{x=2}\\{y=4}\end{array}\right.$

Substituting $x=2$ and $y=4$ into the equation, we get:
$$2 \times 2 + 4 = 4 + 4 = 8$$
Since the left side of the equation ($8$) does not equal the right side ($6$), option C does not satisfy the condition.

Option D: $\left\{\begin{array}{l}{x=3}\\{y=3}\end{array}\right.$

Substituting $x=3$ and $y=3$ into the equation, we get:
$$2 \times 3 + 3 = 6 + 3 = 9$$
Since the left side of the equation ($9$) does not equal the right side ($6$), option D does not satisfy the condition.

Therefore, the correct solution is option A, which satisfies the given binary linear equation $2x+y=6$.

Thus, the answer is: $\boxed{A}$.