How would you approach your onboarding/first 30 days when 'building influence without authority' was highlighted on the job description?
Information is power! Treat all your initial one-on-one calls with stakeholders as a research endeavor.
Ask about their goals and challenges.
Ask what they hope you can collaborate or drive forward.
Get a sense for their communication style and how they process information.
Ask genuine questions to both learn from them and show you respect the expertise they've built up.
Outside of being a sponge and getting to know your colleagues, I would think through at least one "quick win" project you can complete that directly helps your Product Manager partner, in particular.
Here are some ideas for showing value to your PM partner in the first 30 days:
Help with a small research project to influence the product roadmap.
Help with audience sizing to influence prioritization or make it easier for them to build an investment case.
Do a mini product launch or expand product education to drive adoption and engagement.
Conduct an audit of the marketing funnel and suggest both product and marketing solutions to improve it.
Do a competitive analysis that pertains to key strategic questions at-hand.