Praise be to Allah.
What was narrated from the Prophet (peace and blessings of Allah be upon him) concerning staying up at night during the last ten nights of Ramadan is that those nights should be spent in prayer and dhikr.
Charity during Ramadan is better than charity at other times, but we do not know of anything in the Sunnah to indicate that giving charity in the last ten days is better.
But the scholars stated that righteous deeds are better when done at times of virtue, and undoubtedly the last ten nights of Ramadan are better than any other nights, because Laylat al-Qadr is among them, which is better than a thousand months.
Whatever the case, what is prescribed for the Muslim is to give a great deal of charity throughout Ramadan. The Prophet (peace and blessings of Allah be upon him) was the most generous of people and he was at his most generous during Ramadan. Narrated by al-Bukhari, 6; Muslim, 2308
And Allah knows best.