There is no Youth Hostel in Veracruz City but there are many inexpensive hotels or lodgings known as Casa de Huespedes (guest houses).
Last year I started my 7-month bus trip from Montreal to Argentina by staying in Veracruz about a month and a half, to learn some Spanish and have more fun and security on the road.
For the first month, I rented a large room in nearby Boca del Río for 2,000 Pesos a month. It was in a big house at the beach end of the street, in a residential and restaurant area away from the commercial sector. My room was one of three on the second floor (independent entrance to upstairs and to each room.) and was flanked by a huge terrace. The other two rooms only got rented to seasonal workers a few times so the place was very quiet. The price included Cable TV, use of the hand-washing laundry facilities and clothelines, two Queen-size beds and essential furniture, and a good size bathroom.
I bought a 2-burner stove and a bar fridge (for hot chocolate in the morning and beer the rest of the time.) An extremely safe place, with the owner living on the ground floor. When the Gulf Coast was hit by Hurricane Stan, Boca del Rio got more flooded than it already is after a heavy rainfall and it became unbearable. Also, it was soon boring for me since I'm not a beach nut, although I love living by the sea.
I moved into an inexpensive hotel in the Centre of Veracruz, El Real Caribe, for 3,500 Pesos a month. This isn't expensive but there are many hotels that are way cheaper. My room was called The Penthouse, being the last floor or the hotel, with a small circular stairway to reach the roof and even higher, the large square observatory over the city and the Gulf. It had Cable TV also, but I couldn't hear a thing for the bus noise on the street. The main problem was that as nice as the General Manager was, he didn't manage broken fixtures or plumbing. And I had a lot of cleaning to do to make it livable. Here's a picture of one of the two bedrooms.
I brought in my stove and fridge from Boca. It was great to be right at the heart of the action! After only 2 weeks, however, I knew enough Spanish to hit the road to Argentina so a Mexican friend kept my stuff in storage and I left on October 28, 2005. I only returned to Veracruz mid-August this year and found an apartment in a better neighbourhood, much more quiet and clean.
Here they also rent by the month, starting at 5,500 Pesos, fully-equipped and furnished. Watch out when landlord include Cable TV and Internet connection in the monthly rental price. Those are often pirated and totally unreliable. Deal with the Cable company, which is also the ISP. Excellent service at affordable prices. And get into the swing of things on the right foot by being generous with the concierge, even if you never ask her for a thing... she has the magic wand and she knows how to use it!