We will obtain a somewhat better upper bound than $5/4$, using the result that you mention. The method is a small modification of the one that you used.
Break up the integral as $\int_0^{1/2}\sqrt{1+x^3}\,dx + \int_{1/2}^{1}\sqrt{1+x^3}\,dx.$
Look at the first integral. On the interval $[0,1/2]$, we have $x^3 \le 1/8$, so $1+x^3\le 9/8$, and therefore $\sqrt{1+x^3}\le 3/(2\sqrt{2})$. Multiply by $1/2$, the length of the interval. We conclude that the first integral is $\le 3/(4\sqrt{2})$.
Look at the second integral. On the interval $[1/2,1]$, we have $x^3\le 1$, so our function is $\le \sqrt{2}$. Thus the second integral is $\le \sqrt{2}/2$.
Add up. We get $\int_0^1\sqrt{1+x^3}\,dx \le \frac{3}{4\sqrt{2}}+\frac{\sqrt{2}}{2}.$
The calculator shows that the expression on the right is approximately $1.2374369$, and in particular is less than $5/4$. We could get a better upper bound by choosing the "breakpoint" a bit larger than $1/2$, roughly $0.61$. The reason that the breaking up idea worked is that although our function is close to $\sqrt{2}$ near $x=1$, it is quite a bit smaller than $\sqrt{2}$ on the interval $[0,1/2]$.