In Rome your Christian lady would be relatively safe, I believe, up to the reign of Hadrian or even after. The persecution of Nero is now widely believed to have been 'invented' in the 4th century - in Nero's reign hardly anyone knew about christians, and even Trajan seems ignorant about them until receiving a letter concerning their alleged 'Criminal Activities'. I say this not to correct you, but to enable you to give your character a better choice of time scale - for example, setting the story during the Flavian dynasty, as has been suggested.
If your lady wanted to help the poor and yet not compromise the conventions of her class, maybe she could send a slave to identify stricken families in the poorer areas, and perhaps pay for the services of doctors / midwives anonymously? This could open up some interesting plot lines, as the slave relays information between the poor families and the rich lady, who might even try a little evangelism by proxy! I believe at this time, even christians did not have a problem with slavery, and food would not be an issue with the poor, as the corn dole kept most families from starving.