Apartment in Alfama area of Lisbon

Alfama is the last place I ever recommend people stay! There is almost no public transport, the taxis can't even go to many places because the streets are too narrow, there are zillions of stairs and steep hills to climb - and at certain times of the year and in certain areas it would be extremely noisy!
That said, you definitely need to visit Alfama, just not stay there, IMO.

Thank you so much for that advise. Where would you recommend I look? I'm looking for an apartment via VRBO.com rather than a hotel room.

Baiiro Alta and Chiado
AirBnB
For example: https://www.airbnb.com/rooms/766765

I always recommend Chiado, Baixa or Avenida Liberdade. Chiado is the most hilly, but will have the most restaurants close, and you can catch the metro easily to most places in town. Baixa is flat, but there is a scarcity of good restaurants.

We have stayed twice now in the Castelo area of Lisbon, which is really just above the Alfama. The apartment was on the walk up to the Castle. While it can be busy with tourists in the daytime, it was quiet in the evening, and a lovely spot. I frankly found all of Lisbon busy this year and would not pick based just on that. There were several good restaurants close by, and while it is at the top of a hill,so is much of Lisbon. There are also elevators you can find to use.