Change in price and income using the Slutsky equation Quantity bought is given by
$$
q(m,p) = 0.02 m - 2 p
$$
where $q$ is number of bottles bought, $m$ is income and $p$ is price per bottle.

If income is $m = 7500$ and price is $p = 30$, the number of bottles is
$$
q(7500,30) = 0.02 \cdot 7500 - 2 \cdot 30 = 75 \cdot 2 - 60 = 90
$$

Now if price rose to $p = 40$, how much income would I need in order to buy the same amount as before the price change?

I would just solve $q(m,40) = 90$ with respect to $m$, i.e.
$$
q(m,40) = 0.02 \cdot m - 2 \cdot 40 = 90 \Leftrightarrow m = \frac{90 + 2 \cdot 40}{0.02} = 8500
$$

But apparently, I should have used the Slutsky equation, so the answer is something like
$$
\Delta m = q \Delta p = 90 \cdot (40-30) = 900
$$
so the new income is $7500 + 900 = 8400$.

But with an income of $m = 8400$ and price $p = 40$, I cannot afford 90 bottles since $q(8400,40) = 0.02 \times 8400 - 2 \times 40 = 88$, So why is this solution correct?

Why is my solution incorrect and how is $\Delta m = q \Delta p$ related to the Slutsky equation?
The question is not asking what you solved. You literally gave how much income would be needed by buy $90$ bottles at the new price. What the question is asking is basically "calculate the income effect, and add it to the original income, in order to cancel it out".

That is, you solved for what he would have bought if there was no income or substitution effect, but the answer wants to know what he would have bought if there was no income but still a substitution effect.

How the slutsky equation comes into play? That formula you gave is the income effect. The difference of two bottles is the substitution effect.

From here on is an attempt at providing intuition that may be wrong in some aspect. I would not read it unless the above paragraphs did not make sense.

At original prices and income you want to buy $90$ bottles. At the new price and income you want to buy 
$$
.02 \cdot 7500 - 2\cdot 40 = 70\text{ bottles}
$$
you the quantity demanded has changed by $20$. That is,
$$\frac{d x}{dp} = 20
$$
(counting a ten dollar price change as small, which is probably not the best assumption).

The slutsky equation says that we can decompose this change in quantity demanded into an income and substitution effect. That is,
\begin{align}
\Delta x \text{ price change } = \Delta x \text{ substitution effect} + \Delta x \text{ income effect}
\end{align}
Our income has decreased by the number of bottles we were buying times the change in price (This is important to understand, IMO). Think about it, if you were buying $90$ bottles, and the price went up by $\$10$, now you are paying $90\cdot 10 = 900$ dollars more. Therefore, after the price change it is like we are facing prices of $\$30$ and income of $7500-900=\$6600$, if we are discussing the income effect. That is, if we are discussing the idea/thought/alternate reality where, instead of prices changing, we had prices be the same but instead our income decreased by $900$ dollars, we would demand
$$
6600\cdot .02 - 2\cdot 30 = 72
$$
Which means that we have a substitution effect of $2$ and an income effect of $18$. So to cancel the income effect, we would need enough original income to buy $70 +18$ bottles, which is $8400$ dollars.

Now how/why these decompositions work I am still trying to better understand myself, so my reasoning may be off or wrong somewhere. There are also different ways to consider/define the income/substitution effects, which is something to take into consideration.