Start-ups are super exciting, fun and challenging. You have a notch above of all these when you join as a product leader/manager in start-ups where the product offering is unique yet interesting, the potential is high but the road ahead is less travelled and undiscovered.
Your discussions with the founders give you an insight into the current status and what is the expectation going forward:
Additionally, there will be inherent expectations like setting up the process for product management, improve the product line, reduce customer churn rates (attributed to products) and scale the process to build newer products.
I have tried putting together a small check-list of things a product manager/leader should do when they join an early-stage startup. 
The first few weeks/months are going to be tough, but this is when the product manager can help lay down strong foundations from a product management perspective. Below is the checklist which I recommend, in no particular order:
Product Strategy
An understanding of the strategy is the first step, as it helps understand the philosophy of the startup, the founders and the people.
Customer
Knowing the customer and the target market provides insights into the type of customers and how the sales team articulates the value of the product to the customer
Process
The process is going to be one of the most important aspects which the Product Manager will define and manage as part of his role. As a start, he/she should review
Product Content
By leveraging the right content strategy, it is easier to attract potential customers, empower them, thereby converting them to customers. As a first step, to review the content strategy, the PM must understand:
People
This will be a vital element of the entire product management discipline. Product strategy and roadmap will provide necessary inputs for the people element of product management.
Though the above list is not completely sacrosanct, these are some of the aspects of product management, I believe every product manager should look into when they join an early-stage startup.
One of the things that need to be borne in mind is that many of the above elements may not be in place or maybe implemented partially / incorrectly.
So, the onus is on the product leader to build many of these elements ground up. This would be an amazing learning experience for the upcoming product leader.
Hope this list is of help for product managers and leaders starting up at an early stage startup! Let me know your thoughts in the comments.