El Barrio serves breakfast until 1pm which means you can have it for lunch too. There are about 8 magical egg dishes that taste like everything we love about Mexico. My favorite is a couple of over easy eggs on a bed of hash browns with poblanos and cream on top. There is another smothered in mole sauce. This little cafe is right behind the twin houses on Paseo Montejo, it´s open on Sunday too! I would go here for the first breakfast you have in town and if you like it come back and try everything on the menu. Afternoon they serve home backed cakes and pies and pastries.
There are 3 hotels I recommend for breakfast. I think breakfast at Rosas y Xocolate hotel is very good. I like to sit on the inner courtyard terrace and patiently wait for them to screw up my order. If I have the coffee and the juice I generally don´t care how slow they are. Eventually I get what I like and I´m happy to have spent a pleasant morning sending eggs back to the kitchen to be property cooked. Still I would recommend you try the experience for yourself. Also on the Paseo de Montejo you´ll find a little hotel where the Iranian owner had a famous bakery in Mexico City before opening Casa San Angel. Here you can have a good Mexican breakfast like huevos divoricados or rancheros. You can also have a normal hotel breakfast American or continental style. It´s a good place to try chilequiles as well. But as hotels go I like the Hyatt Regency breakfast buffet. They have excellent choices of Mexican dishes that you serve yourself. In the old days I´d have a plate of bacon. You can order your eggs as you like them and they have about 8 freshly made salsa to try. It´s not a place to instagram but the breakfast is great. I didn´t say anything about the lunch.
For those who aren´t wild about breakfast I´d say try one of the new cafe´s around town like Manifesto or Escargot Rustico in Santiago. Caffe Latte Quatro Sette in Santa Ana or Escargot near the new Music Museum on calle 58. They all have good pastry and good coffee. Latte also has great avocado toast. and they are all instagramable.
Platos Rotos is the place to have a variety of dishes you´ll find around Mexico City. It´s in the slow food market on Saturdays or any other day of the week. All of the markets will have a stall where you can sit and have good eggs and tortillas but none of them will have a good cup of coffee or tea. But most would have a good fresh juice. Try El Virgen Morena in Santigao market.