Ongenge bought 3 trousers and 2 shirts for sh. 4600. Okara bought 2 trousers and 3 shirts for sh. 4400. Find the cost of one trouser and one shirt.

Let $t$ be the cost of one trouser.
Let $s$ be the cost of one shirt.

From the problem statement, we can form two linear equations:
Ongenge bought 3 trousers and 2 shirts for sh. 4600:

$$3t + 2s = 4600 \quad (1)$$

Okara bought 2 trousers and 3 shirts for sh. 4400:

$$2t + 3s = 4400 \quad (2)$$

To eliminate $t$, multiply equation (1) by 2 and equation (2) by 3:

$$2 \times (3t + 2s) = 2 \times 4600 \implies 6t + 4s = 9200 \quad (3)$$ $$3 \times (2t + 3s) = 3 \times 4400 \implies 6t + 9s = 13200 \quad (4)$$

$$(6t + 9s) - (6t + 4s) = 13200 - 9200$$ $$5s = 4000$$ $$s = \frac{4000}{5}$$ $$s = 800$$

$$3t + 2(800) = 4600$$ $$3t + 1600 = 4600$$ $$3t = 4600 - 1600$$ $$3t = 3000$$ $$t = \frac{3000}{3}$$ $$t = 1000$$

The cost of one trouser is sh. 1000.
The cost of one shirt is sh. 800.

The cost of one trouser is $\boxed{\text{sh. 1000}}$.
The cost of one shirt is $\boxed{\text{sh. 800}}$.