I am reading the paper "Canonical models of surfaces of general type" by E. Bombieri. In the last section of this paper, there is a statement saying that surfaces with $K^2=1$ and $p_g=0$ do not have pencils of genus $2$, and there is no proof. Is there a proof of this statement?

In fact it seems that the statement is not correct. The paper [Calabri, Ciliberto, Mendes Lopes, Numerical Godeaux surfaces with an involution. Trans. Amer. Math. Soc. 359 (2007), no. 4] contains the classification of numerical Godeaux surfaces (i.e., minimal surfaces of general type with $K^2=1$ and $p_g=0$) that have an automorphism of order 2. The examples described in section 6 have a pencil of curves of genus 2 (cf. Remark 6.3). 


I do not know whether this result is true. In fact G. Xiao in his book Surfaces fibrees en courbes de genre deux, page 66 claims that a surface of general type with $p_g=q=0$, $K^2=1$ and a pencil of curves of genus $2$ has been constructed by Oort and Peters in their paper A Campedelli surface with torsiongroup $\mathbb{Z}/2$. You should check their construction, since unfortunately I have not time to do it now. Notice that Oort and Peter call "Campedelli surface" what is nowadays called "Numerical Godeaux surface" (i.e., a surface of general type with $p_g=q=0$, $K^2=1$); in fact, the name "Campedelli surface" is currently used for surfaces with $p_g=q=0$, $K^2=2$. At any rate, the following is surely true:
See the paper by Catanese and Pignatelli Fibrations of low genus I, Section 5. The proof given there is not based on Bombieri's paper, but it uses a structure theorem for genus $2$ fibrations which involves vector bundles techniques. 

