How can he catch up with the time of prayer?

Firstly: 

The fuqaha’ are unanimously agreed that the one who catches up with a rak‘ah of prayer before the time for it ends has caught up with the prayer in time, but they differed concerning one who catches up with less than a rak‘ah – has he caught up with the prayer in time or not?  

A number of them were of the view that he has caught up with the prayer in time if he says the opening takbeer (before the time for the prayer ends). So if he says the opening takbeer before the time for the prayer ends, then he has caught up with the prayer in time. This is the view of the Hanafis and Hanbalis. 

Others were of the view that he has not caught up with the prayer in time unless he does a complete rak‘ah. This is the view of the Maalikis and Shaafa’is, and it is the more correct view, because the Prophet (peace and blessings of Allah be upon him) said: “Whoever catches up with a rak‘ah of the prayer has caught up with the prayer.” Narrated by al-Bukhaari (580) and Muslim (607). 

And he (peace and blessings of Allah be upon him) said: “Whoever catches up with a rak‘ah of Fajr prayer before the sun rises has caught up with Fajr, and whoever catches up with a rak‘ah of ‘Asr prayer before the sun sets has caught up with ‘Asr.” Narrated by al-Bukhaari (579) and Muslim (608). 

The former (i.e., the Hanafis and Hanbalis) quoted as evidence the report narrated by Abu Hurayrah (may Allah be pleased with him) from the Prophet (peace and blessings of Allah be upon him) who said: “Whoever catches up with a prostration of ‘Asr prayer before the sun sets, let him complete his prayer, and if he catches up with a prostration of Fajr prayer before the sun rises, let him complete his prayer.” Agreed upon. Al-Nasaa’i narrated: “he has caught up with it.” And because with regard to the issue of catching up with prayers, if it has to do with one of the rulings on the prayer, then catching up one rak‘ah or less is the same, such as catching up with congregational prayer and for a traveller catching up with the prayer of a resident, and the first hadeeth may refer to that, but the apparent meaning is more appropriate.  

See: al-Muntaqa by al-Baaji (1/10); Tuhfat al-Muhtaaj (1/434); al-Mughni (1/228); al-Insaaf (1/439). 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: The second view (that of the Shaafa’is and Maalikis) is that one cannot catch up with the prayer except by catching up with a rak‘ah, because the Prophet (peace and blessings of Allah be upon him) said: “Whoever catches up with a rak‘ah of prayer has caught up with the prayer.” This view is the correct one, and it is the view favoured by Shaykh al-Islam Ibn Taymiyah, because this is the apparent meaning of the hadeeth, which is a conditional sentence: “Whoever catches up with a rak‘ah has caught up with…” What is understood from this is that the one who catches up with less than a rak‘ah has not caught up with the prayer. … Other kinds of catching up are also based on this, such as catching up with a prayer in congregation: Does he catch up with the congregation by praying one rak‘ah, or by saying the opening takbeer? 

The correct view is that he cannot catch up with it except by praying a rak‘ah, just as he cannot catch up with Jumu’ah prayer except by catching up with a rak‘ah, according to scholarly consensus. End quote from al-Sharh al-Mumti’ (1/121). 

As you had prayed the first rak‘ah of Zuhr before the adhaan was given for ‘Asr, you caught up with the prayer at the time when it was due. 

Secondly: 

The sleeper is excused for the time when he is asleep, but when he wakes up it is obligatory for him to pray. It was narrated that Anas ibn Maalik (may Allah be pleased with him) said: The Prophet of Allah (peace and blessings of Allah be upon him) said: “Whoever forgets a prayer or sleeps and misses it, the expiation is to offer the prayer when he remembers it.” Narrated by al-Bukhaari (572) and Muslim (684). 

And he (peace and blessings of Allah be upon him) said: “There is no negligence in sleep, rather negligence is the fault of one who does not pray until the time for the next prayer comes. Whoever does that, let him pray when he remembers it.” Narrated by Muslim (681). 

And Allah knows best.