There is a shortage of budget accommodations in Beirut and it doesn't look like that's going to be changing anytime soon with all the glitsy luxury hotels under construction such as the Four Seasons and Grand Hyatt.
I can recommend the Casa d'Or Hotel and the Napoleon Hotel. Both are located in the laid back Hamra district of the city and is close to the coast, AUB, and downtown area. I've had relatives stay at both of these hotels and they've been pleased with their stays.
Any of the chain-based hotels will be considerably more expensive than most of the non-chain hotels (except for the boutique hotels). Even the Holiday Inn you'll find to be quite expensive.
As for hidden gems, Beirut itself is a fascinating, fun, and friendly city with a lot of things to do. Definitely walk the corniche from the Raouche Rocks to the iconic InterContinental Phoenicia Hotel, it's a beautiful walk, especially at sunset. Avoid it on Sundays however as it gets unbelievably crowded. The beautifully restored historic downtown is a must see with all its beautiful architecture and many shops, restaurants, and cafes. The Gemmayze neighborhood just east of downtown is worth seeing and is a major nightlife spot. The trendy Verdun street (Rachid Karame Street) is also pretty nice place, especially in the evening. For some shopping, Hamra street and the surrounding area is great place. Hamra Street is ideal if you are shopping for local souvenirs. Besides shopping, there are a lot of cafes and fast food outlets. If you stay at one of the two hotels I recommended, you'll be right in the heart of this neighborhood.
Lebanon is famous for its cuisine and I would say the best Lebanese food in town is at Karam located in downtown. Everything from the tabbouleh salad to the kafta grill is outstanding. It's one of the more expensive places but very well worth it. For fast food, I highly recommend BarBar in Hamra street. There are several BarBars in Beirut, but they are not all the same, the one in Hamra is the best in my experience. Their shawarma and falafel are amazing.
If you have time, I would definitely venture out of the city. Either negotiate a day taxi or take an organized tour. For organized tours, I highly recommend Nakhal, they're very professional and feed you at excellent restaurants.
Let me know how long you'll be staying in Beirut and I can make some recommendations on excursions.
Now for Damascus, I'm not an expert as I've just visited it once as a day trip from Beirut. The historic old part of the city is absolutely fascinating and quite extensive. All the important sites such as the various Khans, Souk al-Hamideye, Azem Palace, Tomb of Saladin, Umayyad Mosque, and Damascus Citadel are all very close to each other. For food, I highly recommend Al Khawali Restaurant located in a restored old house near the Azem Palace. The food is excellent and the prices are extremely reasonable. You will find Damascus to be much cheaper than Beirut. I highly recommend you check out the Al-Nawfara cafe which is a 300-year old continuously operating coffee shop. It's close to the Umayyad Mosque.
Plan at least 3 hours traveling time from Beirut to Damascus, it may take slightly less, it may take a bit more, depending on the traffic and the border crossing. Keep in mind that you'll have to go through passport control and customs on both the Lebanese and Syrian sides of the border.
Sorry, I do not know anything about accommodations in Damascus, although you should be able to find reasonable rates quite easily as it is quite an affordable city.
"Generosity is giving more than you can, and pride is taking less than you need." - Khalil Gibran