You are to bear in mind the fact that fasting is one of the basic pillars of Islam. The Prophet (peace and blessings be upon him) said, “Islam is built upon five pillars: testifying that there is no true god except Allah and that Muhammad is the Messenger of Allah, performing Prayer, paying the Zakah, making the pilgrimage to the Sacred House (Hajj), and fasting the month of Ramadan.” (Reported by Al-Bukhari)
Regarding your question, a Muslim is encouraged to hurry to make up for the missed fast days of Ramadan. It is allowed to make up for the missed fast days of Ramadan in any month that comes after Ramadan. However, Sha`ban may be the last chance for making up for the missed fast days of the past Ramadan before the coming of other Ramadan.
Responding to this in more details, the eminent Muslim scholar Sheikh Yusuf Al-Qaradawi states:
A Muslim male or female should make up for the missed obligatory days of Ramadan whenever he or she can. The missed fast days are to be offered at any time of the year before the coming of the next Ramadan. This implies that there are eleven months for a Muslim to make up for the missed fast days, whether the obligatory fast of Ramadan was broken due to illness, travel, menses, or any other reasons.
There is flexibility in Islamic Sharia regarding when one is permitted to make up for the missed fasting days. One can make up for the missed fast days in Shawwal, that is, right after Ramadan, and also in the months that follow.
Undoubtedly, it is advisable that one speeds up in making up for the missed fast days. Allah says about hastening in doing good: (… so vie with one another in good works) (Al-Ma’idah 5: 48).
It is better for one to make up for missed days because nobody knows when his end will come. In case one delays making up for the missed fast days, whether because of hot climate, weakness, or a physical ailment, or because one is preoccupied with certain duties preventing one from observing fast, then one has time until the next Ramadan.
When Sha`ban comes, one should make up for the missed days, for it is one’s last chance. It is to be noted that `A’ishah, the Mother of the Believers (may Allah be pleased with her), used to make up for the missed days of Ramadan in the month of Sha`ban.
Consequently, it is allowed to make up for missed fast days in Sha`ban. Those who have doubts about that should know that there is nothing that renders this unlawful in the Sharia because it is allowed to make up for the missed fast days during any month.
Having clarified the above, I should add here that making up for missed fast days of Ramadan can be delayed and performed later whenever a person is able; this is in cases when the person remains sick until the next Ramadan. Hence, making up for the fast days will be like a debt that one should pay whenever one becomes able. Doing so is permissible, where Allah says by the end of a Qur’anic verse about fasting: (Allah desires for you ease; He desires not hardship for you) (Al-Baqarah 2: 185).