Should he sponsor orphans or the elderly?

Sponsoring the elderly and relieving them of harm is a communal obligation (fard kifaayah). They should be provided with a life of dignity and they must be given all human rights in full. Undoubtedly there is much good in doing that to seek the pleasure of Allaah, and whatever you spend on their care will be recorded with multiple reward, in sha Allaah. Allaah says (interpretation of the meaning): 

“The likeness of those who spend their wealth in the way of Allaah, is as the likeness of a grain (of corn); it grows seven ears, and each ear has a hundred grains. Allaah gives manifold increase to whom He wills. And Allaah is All-Sufficient for His creatures’ needs, All-Knower”

[al-Baqarah 2:261]

Moreover, by doing this you are relieving distress and hardship, and helping a Muslim brother, and the Messenger of Allaah (peace and blessings of Allaah be upon him) encouraged that when he said: “Whoever removes a worldly hardship from a believer, Allaah will remove one of the hardships of the Day of Resurrection from him. Whoever grants respite to (a debtor) who is in difficulty, Allaah will grant him relief in this world and in the Hereafter. Whoever conceals (the fault of) a Muslim in this world, Allaah will conceal him (his faults) in this world and in the Hereafter. Allaah will help a person so long as he is helping his brother.”

Narrated by Muslim (4867).   

Undoubtedly the reward for sponsoring an orphan is great, but a person should nevertheless give priority in charitable giving to those whose poverty and need are greater. An orphan may have wealth from which he may spend, or he may have a rich person who spends on him, so someone else, who has no wealth and no one to spend on him, may be more deserving of charity and help, and that may bring more reward, in sha Allaah, than spending on an orphan. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said, after mentioning the eight types of people who are entitled to zakaah: 

If it is said: who is most deserving of zakaah funds out of these eight categories? 

We say: The one who is most deserving is the one whose need is greatest, because all of them meet that description, so the one whose need is greatest is most entitled to it. In most cases the ones who are in greatest need are the poor and needy, hence Allaah started with them when He said (interpretation of the meaning): “As-Sadaqaat (here it means Zakaah) are only for the Fuqaraa’ (poor), and Al‑Masaakeen (the poor)” [al-Tawbah 9:60]. End quote. 

Majmoo’ Fataawa Ibn ‘Uthaymeen (18/339). 

And Allaah knows best.