The problems seem much deeper than insufficient financial means. Remember that any of the reasons provided is enough to refuse the visa, the fact that the consular officer checked several boxes means that you have several issues to address and it will be all the more difficult to reverse this decision in the short-term. In particular, they deemed that
- The information provided regarding the justification for the purpose and conditions of the intended stay was not reliable.
In other words, they do not believe your sister's intentions regarding her trip to the Schengen area and why she wants to go there. They might even suspect some of the documents you submitted are not genuine. You can read some info about that in Schengen Visa Refusal: Justification for the purpose and conditions of the intended stay was not reliable but in general it's a pretty broad reason and one that's difficult to overcome for a private trip.
- There are reasonable doubts as to the reliability of the statements made as regards on your economic ties to home country as well as on the origins of the financial means.
This goes into the financial means aspect. You already submitted your bank statements and evidence of your income so there is nothing obvious you could do to support her application. Instead, the refusal seems to be about your sister's financial means and her ties to the country, basically whether she is wealthy or not. You can pay for her trip if you want but it doesn't really matter, that's not what the consulate cares about.
You said you would “sponsor” her and that's a term we read a lot but that's not really the way it's supposed to work in the Schengen area. Sponsorship is only for visits to people who reside in the Schengen area and exempts you from booking a hotel. It doesn't replace showing that your sister could afford the trip on her own and has strong reasons to come back to Pakistan.
From your comments, it seems you already submitted evidence of her job and income so apparently this job wasn't enough to convince the consulate. In fact, all this could even have worked against her as, unlike spouses, adult siblings aren't necessarily expected to pay for each other's expenses. If she needs to rely on your money or maybe has family in Europe, it makes her look like someone who risks trying to stay in the Schengen area (illegally) after their visa has ended.
- There are reasonable doubts as to your intention to leave the territory of the Member States before the expiry of the visa.
That may be the most difficult reason to overcome. The consular officer explicitly draws the conclusion I mentioned earlier. Practical ways to overcome this is to show a history of travel to other countries, preferably richer countries or maybe having a husband and kids but if that's not the case for your sister, there is very little she can do to get a visa.
It's also a bit of a catch-22, as if you appeal and apply repeatedly, you will appear desperate to go to Austria, which agains calls your intent into question. As unfair as it is, there are many people who simply cannot get a Schengen visa because of their personal situation (age, wealth, etc.)
Could the calls have influenced the decision?
Maybe, depending on what you said, but it's hard to believe that it would have been decisive. Conversely, it's very unlikely a call could help in any way. The refusal rate in Islamabad is high and this sounds like an open-and-shut case from the perspective of the consular officer, not something that was decided on some small incidental information.
What should I focus on when submitting an appeal to ensure a better outcome this time?
I am not familiar enough with Austrian law to know if an appeal is the right procedure but it might already be doomed on purely procedural grounds. Even if you can submit new evidence, the refusal seems to be based on your sister's personal situation and this will always be difficult to overcome in the short-term.
There is a lot of information you didn't provide and in general I think appeals are best left to lawyers but my main advice at this point is to take time to consider everything that's been mentioned in this Q&A and avoid getting too much hopes that your sister will be able to visit Austria in the short-term.