Is measure preserving function almost surjective?

Yes, by Luzin's theorem. Fix $\varepsilon>0$ and take a compact subset $K$ of measure at least $1-\varepsilon$ such that $F$ is continuous on $K$. Then $F(K)$ is a compact set of at least the same measure as $K$, since $F^{-1}(F(K))\supset K$. So, for any $\varepsilon>0$, $F([0,1])$ contains a measurable subset of measure $\geqslant 1-\varepsilon$. This implies that inner measure of $F([0,1])$ is 1.

A few remarks (where $m$ is the Lebesgue measure):

1. If $F[0,1]$ is a Lebesgue measurable set, then $F[0,1]$ has Lebesgue measure equal to one (you can read Fedor Petrov's comment for this post).
2. Because of Birkhoff's Ergodic Theorem: For every $[a,b]\subset [0,1]$ $$ \int_0^1\left(\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n} 1_{[a,b]}(F^i (x))\right)dm=m([a,b]). $$
3. If $F[0,1]$ is a Lebesgue measurable set and $F$ is also Ergodic, then for $m$-a.e. $x$ $$ 1=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n} 1_{F[0,1]}(F^i (x))=m(F[0,1]), $$ which is another proof that $m(F[0,1])=1.$
4. If $F[0,1]$ is a Lebesgue measurable set, then $$ 1=\int_0^1\left(\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n} 1_{F[0,1]}(F^i (x))\right)dm=m(F[0,1]), $$ which is another proof that $m(F[0,1])=1.$
5. If $T$ is ergodic and $K\subset [0,1]\setminus F[0,1]$ is a Lebesgue measurable set, then $m(K)=0.$ By contradiction, we would obtain that for $m$-a.e. $x$ $$ 0=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n} 1_{K}(F^i (x))=\mathcal{Leb}(K)\neq 0. $$
6. If $K\subset [0,1]\setminus F[0,1]$ is a Lebesgue measurable set, then $m(K)=0.$ By contradiction, we would obtain that $$ 0=\int_0^1\left(\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n} 1_{K}(F^i (x))\right)d m=m(K)\neq 0. $$
7. Let $\mathcal{A}=\{K:K\subset F[0,1], K \mbox{ measurable }\}$ and $\mathcal{B}=\{K:K\subset [0,1]\setminus F[0,1], K \mbox{ measurable }\}.$ Then $\sup_{K\in \mathcal{A}} m(K)=1.$ By contradiction, if not, by sigma aditivity $$ 1=m[0,1]=\sup_{K\in \mathcal{A}} m(K)+\sup_{K\in \mathcal{B}} m(K)=\sup_{K\in \mathcal{A}} m(K)<1, $$ which is another proof that the image of $F$ has inner measure one.