Total numbers with $4$ as their unit place : $10$ $\{4,14,24,34,44,54,64,74,84,94 \}$
Total numbers with $4$ as their tens' place :$10$ $\{40,41,42,43,44,45,46,47,48,49 \}$
Sum of $100$ numbers without updation : $\Large \frac{100 \times 101}{2} $ $=5050$
Every unit place will increment the sum by $1$ and tens' place by $10$.
$\therefore $ Total incremented value = $10 \times 10 + 10 \times 1 =110$
Complete Sum = $\color{Red}{5160}$
Footer note :
$44$ is special case here so if i mentioned in both set,though it can be tackled separately but gist will be same like :
$\Rightarrow $ First treat it as Row element of $40$ then increment is as $\color{Red}54$
$\Rightarrow $ Now as unit place element then increment is as $\color{Red}{55}$
But overall change is $55-44=11$ same like above calculation.