Did Monk's mother also suffer from OCD?

No, but she was a control freak.

Agnes Monk is a supporting character—God bless her heart—that didn't even make it into TV Tropes' Minor Characters list. Regardless, what little we know about her is precious and sufficient.

She [Agnes] was also something of a control freak, telling Adrian not to share his school lunch with anyone and to sit near the fire doors. She made her sons practice laughing to make them seem normal but did not succeed.

— TV Tropes

According to CED,

Control freak—someone who is determined to make things happen in exactly the way they want and who tries to make other people do what they want

As a control freak, she would have had an opinion about how things should be and enforced it. Thus, it must have been her own idea to label the mugs to instil discipline in the boys from a green age.

Though, as always, we prefer to end on a bright note; Agnes Monk was not all gloom and doom:

He [Adrian] has complicated feelings towards his mother, because while she was a paradoxically fussy and distant maternal figure, she never stopped believing in Adrian and tragically died before she could see him come into his own as a police officer.³

To conclude, it was probably her idea to number the mugs because she was a control freak.

It is strongly implied, but not definitively proven.

00:35:07 Mother numbered
00:35:09 So we'd always put them back in the right order.

This is typical OCD behavior. She had a preferred order, and the boys didn't respect that order, so she made it clear using numbering.

But, more importantly for your question:

00:35:11 Of course she did.

This established that this kind of behavior is to be expected of their mother, i.e. that she had many similar behaviors. This strongly suggests that she had OCD tendencies, but it's not enough for an ironclad diagnosis.

or was Adrian's and Ambrose's need for order such that she numbered the mugs so that her two sons wouldn't have to agonize about which mug went where?

Firstly, this doesn't make logical sense. If she numbered the cups, that means that without the numbers, the boys couldn't tell the cups apart. Because if they could do it without the numbering, the numbering would not have been necessary.

If they couldn't tell the cups apart without being numbered, they couldn't be agonizing over the "wrong" cup order to begin with.

Back when the cups weren't numbered yet, the alleged cause of the numbering (i.e. the boys being the ones who were upset about the wrong cup order) couldn't possibly exist, since the transcript makes it clear that the numbering was done to make the boys able to put the cups in the right order, which implies that they weren't able to do it before numbering the cups.

Secondly, that's not what the transcript tells us:

00:35:07 Mother numbered
00:35:09 So we'd always put them back in the right order

This states the intent of the numbering, rather than the reason for it. The ability to put the cups in the right order is the effect, not the cause, of the numbering.

Ambrose had been living in that house for 32 years before this episode so it´s more plausible to understand that he established the order of the cups or more probably both brothers did when they were younger.

It´s true that the mother was supposed to have some mental issues as she was 'catatonic' (perhaps and exaggeration) for some time after their father left them.

No, Adrian developed OCD at a young age, which meant that he was the one who pu the numbers on the cups