You can prove the inequality by using finite additivity and monotonicity; by monotonicity we mean that if $A\subset B$ , then $m^*(A)\leq m^*(B)$, and additivity means that if $A,B$ are disjoint, i.e., if $A\cap B$ is empty, them $m^*(A\cup B)=m^*(A)+ m^*(B)$. Then, given any $A,B$, you can "disjointize them" , i.e., you can find a pair of sets A',B' so that A',B' are disjoint, and A\cup B=A'\cup B', (what we want is A'.B' disjoint., so that m^*(A\cup B)=m^*(A'\cup B').)
Then the first , i.e., leftmost part of the inequality, $m^*(A)\leq m^*(A\cup B)$, follows by additivity of the measure, since $A\subset (A\cup B))$ (sorry, I don't know how to do non-strict inclusion ).
For the other part of the inequality, i.e., to show that $m^*(A\cup B) \leq m^*(A)+m^*(B) $ we can define sets A',B' as above, by just setting :$A':=A$, and B':=B-A , where '-' just means set complement. Then, since A',B' are disjoint, and A\cup B=A'\cup B', we have that m^*(A\cup B)=m^*(A'\cup B')=m^*(A')+m^*(B'):=m^*(A)+m^*(B-A)
We then have, by transitivity, that $m^*(A\cup B) \leq m^*(A)+m^*(B-A)$. (##)
Now, use the fact that $(B-A)\subset B$, so that, by monotonicity $m^*(B-A)\leq m^*(B)$ and we substitute this in (##) above , so that m^*(A'\cup B')=m^*(A')+m^*(B')=m^*(A)+m^*(B-A)\leq m^*(A)+m^*(B-A) \leq m^*(A)+m^*(B), and we are done.