For cheap meals in the old city, visit the Arab/Muslim quarter for falafals.  Or you can get a good thick slice of pizza in the Armenian quarter.  Remember that the Jewish quarter shuts down completely for the sabbath, but the little grocery store there has things like yogurt and bread and cheese that you can make a nice picnic out of. 

Beware as elsewhere of tourist traps.  Jerusalem has some very fancy and expensive restaurants. An inexpensive buffet lunch is available at noon each day at the English Church opposite the Citadel. Ten minutes walk west from the Jaffa Gate brings you to the restaurant section of Nahlaot along Jaffa Road where you can find places open all week -including Saturdays.

Somewhat out of the way for most tourists, the large shopping mall at Malha(Manahat) near the railway station, opposite the football stadium  in SW Jerusalem has a varied "food court" with dairy , vegetarian and meat meals.  The food court in the Central Bus Station, which is also a shopping mall on Jaffa Street, is much more accessible. Closed Friday evening to Saturday evening.

Other inexpensive cafeterias at the University campuses :Givat Ram and Mt Scopus are open to all. Closed Saturdays.

 For tasty inexpensive eats and colourful sights try the Mahane Yehuda Market in west Jerusalem and sample the fresh fruit ,and bakery products.Open all week except Saturday. Buses that run along Aggripas Street  stop  there, but best approached via the Jerusalem Light Rail which runs the length of Jaffa Street.