In really near suburbs you can find pretty cheap but still big enough apartments.
Ain el Remmaneh
for example is 5 or 10 minutes away from Ashrafieh
, depending on where in this area you live (if there is a lot of traffic, it could take you 30 or 40 minutes depending on where you want go in Ashrafieh
cause it is a big area).
Other than Ain el Remaneh, you have Badaro (slightly more expensive), Furn el Chebbak, Sin el Fil and Ras el Nabeh/Sodeco.
Those are the main areas surrounding Ashrafieh that are not considered to be inside the city but very very close suburbs with flats that are quite cheap
(pour te donner un petit exemple, tu peux facilement trouver un bon T1 bis ou T2 meublés à 300 euros)