Interesting pattern...

Consider a 10x10 grid containing values 0 to 99, with 00 in the upper left corner, and 99 in the bottom right. Each row contains values with the same tens digit, Each column contains values with the same ones digit. Notice, $0+1+2+3+4+5+6+7+8+9 = 45$ .
Notice there is symmetry in that the sum of the ones digits across any row is always 45, and the sum of the tens digits across any column is also 45. Taking the sums separately gives $45\times10 + 45\times10 =2\times45\times10= 900$ since 100 was left out of this grid you may add one to account for its absence. giving $901$

Let $d_{n|m}$ denote the sum of digits of number from $n$ to $m$ .We can notice that the sum of digits of the number follows in following pattern.

$\begin{aligned} d_{0|10} & = 9\cdot 0 + 45 + 1 \\ d_{11|20} & = 9\cdot 1 + 45 +2 \\ d_{21|30} & = 9\cdot 2 + 45 + 3\\ \vdots \ & = \quad \vdots \qquad \vdots \\ d_{90|100} & = 9\cdot 9 +45 + 1\phantom{ccc}\text{since}{\color{#3D99F6} 1=1+0+0 } \\ \hline d_{0|100} & =9\sum_{j=1}^{9} j + 45\cdot 10 + \sum_{j=0}^{9}j +1\\ & = 9\cdot 45 +45\cdot 10 +45+1\\ & = 45\,(9+10+1) +1 = \boxed{901}\end{aligned}$

Let $S_n$ denote the sum of the first $n$ digits. Observe that: $11 + 12 + ... 19 = (10 +1) + (10 + 2) + ... + (10 + 9)$

Hence, we compute $S_9$ 10 times, and adding each multiple of 10, 10 times. So we get the following equation: $10 * S_9 + (10*10 + 10*20... + 10*90) = 450 + 100(S_9)= 450 + 450 = 900$ Considering the 1 in 100, we get $900 + 1 = 901$ .

Every digit $d \in \{0,1,2,3,4,5,6,7,8,9\}$ appears in the numbers $00\text{-}99$ equally often (note that I write the leading zero if necessary, so it is $06$ , not $6$ , but this doesn't change the digit sum). Since there are $100$ numbers, each with $2$ digits, and there are $10$ choices for $d$ , we see that every $d$ appears $\frac{100\cdot 2}{10} = 20$ times.

Therefore the sum of the digits $00\text{-}99$ , or equivalently $1\text{-}99$ , is $20 \cdot (0+1+2+3+4+5+6+7+8+9) = 20 \cdot 45 = 900$

Then just add the digit sum from the number $100$ to get an answer of $900 + 1 + 0 + 0 = \boxed{901}$