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I was reading L’Etranger by Albert Camus and came upon this sentence:
Quand elle était à la maison, maman passait son temps à me suivre des yeux en silence.
The usage of à I am referring to is in bold.
I reckon it means “by”.
Also, sometimes, de is also used to mean by. When to use à and de respectively when translating by.
Rather than trying to translate the word "à", it might be better to understand the phrase as a whole.
The phrase passer son temps means "to spend a lot of time, and even much more than one should reasonably expect". There is a nuance of reproach or complaint, so it differs from the neutral phrase passer du temps.
Mes enfants passent leur temps sur leur téléphone
You can use this phrase with a verb to specify "what" one spends too much time on. As often in French, this is done by adding the preposition à with an infinitive.
J'en ai marre, je passe mon temps à nettoyer la maison
As to your question when to use à or de in such cases (I presume with an infinitive form), I am afraid there is no easy answer: that is something that native speakers learn naturally, but that you, as non-native, will unfortunately need to study... There are some trends you can remember (see here) but for a lot of adjectives, nouns or idiomatic phrases, you just need to learn it case by case.
