He vowed to Allah that he would remember Him for three continuous months

Answer: What appears to me is that this covenant is in the ruling of an oath; for making a covenant with Allah to do such-and-such is a promise reinforced with the utmost degree of emphasis, which here is the covenant; and likewise an oath is, in such a case, a promise reinforced with the utmost degree of emphasis, which is swearing by Allah, majestic and exalted.
A believer ought to fulfill what he has sworn to do of obedience to Allah, Exalted is He. If what he has sworn to do becomes difficult for him or causes him hardship, then he should offer the expiation for his oath, in accordance with His saying, Exalted is He: “Allah has already ordained for you [a way] to dissolve your oaths.” [At-Taḥrīm:2]
And the verse of expiation in Sūrat al-Māʾidah was revealed regarding some of the Companions who had sworn that they would fast and never break their fast, and stand in prayer at night and never sleep, and so on.
As for what came in the censure of Thaʿlabah, that was because he was miserly with the obligatory charity, rebelled against paying it, and did not fulfill his firmly emphasized promise in that regard. He had made a covenant with Allah that if Allah granted him wealth, he would indeed pay Allah’s due in it and would be among the righteous who pay the charities due on their wealth and are not miserly with them.
What is mentioned in the question is not of the category of nadhr (vow); for an act of worship vowed becomes obligatory by the vow when its genus is already obligatory – unlike when a person swears an oath or makes a covenant with Allah to perform it. In that case, the act of worship does not become obligatory by virtue of that; rather, what becomes obligatory is guarding the oath, either by fulfilling it or by performing the expiation. So the legal responsibility here is attached to the oath, whereas in the vow it is attached to the vowed act itself.
If the one who has sworn wishes to fulfill what he swore to do of remembering Allah, Exalted is He, for three continuous months, then if he has a specific intention, the oath is directed by that intention. If he does not have a specific intention, then from the three months there are to be excepted whatever times necessity and need call for – such as the times of sleep, eating, drinking, and the needs of himself and his dependants in seeking a livelihood.
What I see for this person who swore is that he should offer the expiation for his oath, along with fulfilling the following:
 He should perform the five daily prayers together with their regular Sunnah prayers.
 He should glorify Allah, praise Him, and remember Him after the Fajr prayer until sunrise.
 Then he should go to his home and family for breakfast with coffee, according to his habit; and whenever an opportunity arises for him at that time to remember Allah, he should remember Him with praise, glorification, saying “Allāhu akbar,” and the like.
 Then he should go to seek his livelihood, if he is among those who seek a livelihood, and he should seek it as he customarily does, without being heedless of remembrance from time to time as opportunity allows, and without being completely heedless of the remembrance of Allah. When he finishes his work, he should go to the prayer, then to his home for lunch, and do as we mentioned concerning breakfast.
He should go at the end of the day to the gatherings of the seekers of knowledge until the time of Maghrib. After the two prayers (Maghrib and ʿIshāʾ), he should go to his home for dinner, sit with his family, and intend by his sitting with them and conversing with them that which Allah, Exalted is He, has commanded of kind treatment toward them. Then he should go to his bed and remember Allah until sleep overcomes him, and so on. The expiation will cover what he may have been heedless of in the remembrance.

Source: Min Thimār al-ʿIlm wa al-Ḥikmah vol.2