Step 1: Understand the idiom.
The phrase "to pull strings" is an idiom. It refers to using one's connections or influence, often secretively, to make something happen. The imagery comes from a puppeteer pulling strings to control a puppet.
Step 2: Apply the meaning to the context.
In the sentence, the manager needed to get invitation passes, which were likely hard to obtain. By "pulling strings," the manager used his special connections or influence to acquire them.
Step 3: Match the meaning to the options.
"Use personal influence" is the precise definition of the idiom "pull strings".