Learn Dutch: De oude monteur is bezig met de krik. - The old mechanic is busy with the jack.

The phrase bezig zijn met translates to "to be busy with" and is used to show a continuous, task-focused action. Here, it is followed directly by a noun (de krik), which is the simplest way to use this structure. It shows that the mechanic is actively engaged with that specific object right now.

Could I use the continuous aan het structure for this sentence instead?

You couldn't use it directly with just the noun. The aan het continuous structure requires an infinitive verb (for example, is de auto aan het repareren - is repairing the car). Because you are expressing focus on a physical object (de krik) rather than a verb, is bezig met is the required choice.

Why does the adjective oud get an -e at the end to become oude?

Monteur is a de-word, and we are using the definite article (de). In Dutch, adjectives coming before a noun almost always get an -e ending (de oude monteur). As a quick refresher, the main exception is when you have a singular het-word preceded by an indefinite article like een.

What exactly is a krik?