Galicia is the toughest front for the central powers at the start of the game. Not only the Lemberg area, but you also must be careful not to lose Tarnow too quickly because if the Russians get into the hills/mountains they can use their superior body count to push a couple more hexes and cut the rails to Sambor/Stanislaw and then you might lose your entire army.
The Russians have the potential to really fill every hex full of corps in one sub-area to the point that you really can't afford to fight them even with good odds. You have to delay their advance for as many turns as possible until more units become available.
You have to be extremely careful where you place your starting defensive units. One hex too far forward and it's subject to a beatdown (losing a single extra Austrian corps early is a catastrophe). One hex too far backwards and it lets the Russians facehug-push you too quickly.
To be clear, I always lose Lemberg. You can't save this city against a human opponent unless you rail in a German army quite early and in that case you lose out somewhere else. Lemberg falls either from a direct assault or due to losing adjacent hexes. Those nearby hexes have no defensive bonuses and are impossible to defend if the Russians choose to throw the kitchen sink at it. Counterattacks can be very risky and counterproductive even if you do more damage than you take. The Russians have better troops than you (more tech) and better generals too (they should pay for these).
I try to at least delay it a bit and force the Russians to make an attack which is costly. And then after that I think the Russians can take Stanislaw before they get artillery, but they definitely can after they get their artillery working. In this game it's impossible to defend a single particular hex because artillery can always focus it.
The mountains in the southeast is a weird area which I need to explore further. Depending on where you place your starting units the Russians can even threaten one of your mines down there if they rush a cavalry. This can be a problem area that forces you to rail in some extra units. I'm not sure if it's the optimal place for Russia to push in general, but it depends on what units the Centrals railed where.
Part of the game is figuring out where each side is truly massing their resources, and having an idea of how many units they have to work with in total. Pay attention to the report charts which show how much your enemies spent on tech. If you are outspending them on technology, they are probably attempting to maximize pressure on you through a push. If Russia is keeping pace with German tech, then they are turtling and you can be more aggressive.
You want to pay particular attention to the locations of Russian cavalry. These units have the potential to dash quite far into your territory and allow infantry to force march to form a solid wall which can be overwhelming. You need to have reliable ZOC movement penalties in their way to prevent deep horse rides.
All that being said, Austrian morale is fine in the long term even if you lose Lemberg and Stanislaw. You can even lose Przmesyl if you have to. If you have the choice between abandoning Lemberg or risk endangering your entire army, give up Lemberg. Eventually Austria will get plenty of +NM events (Serbia falling, Italy fighting, capturing Italian towns). Defending Lemberg and Stanislaw is not worth delaying the destruction of Serbia.