A well-informed Russian tells me:
'Now Russia has a PMTO (“point of material-technical supply”) in the Syrian port of Tartus inherited from the USSR times, and nothing more is planned for the near future. This PMTO includes one berth (the second has decayed by the present), two depots and one residential building (built in 1988 ) with parking. As they were in bad state, the repair has begun in 2005 and also dredging was ordered, because now the depth near the berth is too shallow for cruiser-like ships.
Thus, the Tartus “base” can be used only for short rest and minimum resupply of little navy squadrons cruising in the Mediterranean or sailing towards the Indian Ocean in order to avoid constraining diplomatic formalities and excessive expenses. The facility is useful and suitable, but it should not be exaggerated.
The main Syrian port of Latakia is actually modernized by the Russian firm now, but these arrangements (mostly dredging) are quite civil and bear no relation to the Russian Navy.'
Judging from the description (i.e. one "berth," with a second one decayed), it sounds like he's most likely describing the two little piers attached to the breakwater at the extreme left of your screenshot. They do seem to lead to some kind of parking lot, but the only buildings around seem to be the ones near that Osa missile boat hauled up on dry land. Maybe that one's Russian too? There seem to be a lot of other patrol boats nearby that are obviously Syrian Navy though, and a sort of half-wall separating the buildings from the piers/parking.
There does seem to be one floating vessel parked at the "berth" though. I don't suppose you can ID it? Maybe the Russians were paying a visit, the day this photo was taken.
-SK