How to wear multiple hats at a SWE job?

First off, congratulations. That is no mean achievement in this economy.
If you're looking to start up, then in all likelihood your first few months (and even perhaps years) will necessitate that you have a great understanding of the business you plan to run. So, in that case, it is a question of just getting your feet wet if and when you want to.

Alternatively, if you prefer to move to a TPM role, here are my two cents.

I think a good TPM has a combination of three offerings -
1. good technical skills (plain-speak - has written code in the past)
2. great people skills (high empathy for engineers, good understanding of business requirements and an even better translation of business needs into the coding world)
3. supreme prescience (being able to have the right pulses to 'know' the future direction of every single initiative)

You probably have the first skill above. If you're still new to the tech world, a few months in a fast-paced unicorn will help you polish your tech skills. Be sure to check this course out - https://www.jointaro.com/course/level-up-your-code-quality-as-a-software-engineer/what-actually-makes-code-good/. It is probably the best lottery ticket out there to learn how to code 'better'.

The second skill is highly experiential. The more time you spend talking to engineers and people with business needs, the better. Much of communication is also about asking questions, collating information, and having a 0->1 understanding of every system you work with.

The third skill is the toughest in my opinion. Two resources come to mind. A - This essay by Paul Graham - https://paulgraham.com/greatwork.html and this remarkable video by @Alex - https://www.jointaro.com/course/maximize-your-productivity-as-a-software-engineer/finding-the-best-things-to-work-on/. Be sure to check out the matrix that he refers to. Doing great work helps you know what is important. And knowing what is important enables you to predict future direction better!

As for timelines, I do think you should do it at a time when your manager thinks you have contributed sufficiently and you have learnt enough of the above three skills to move.

Your manager probably has certain expectations of you fulfilling SWE duties since you recently joined, so I would focus on doing an amazing job there to build trust and prove your worth.

Being a really good SWE will also help you to become a really good PM because it gives you a better understanding of how a tech company operates from a SWE point of view.

As far as timeline, it can take a few months for a new grad to really onboard and feel comfortable in their new role. You really want to show that you can lead a project, not just on the technical side, but you want to be involved in the needs finding phase, design, and project management side as well. This will also help you build trust outside of the team, which will make the conversations easier because your manager is probably going to have side conversations with the people you worked with to see if there is a possibility of you switching roles.

You should basically already be doing PM roles before you ask because it takes time to train someone in a new role, so you want to show that you are already at a certain level or have potential to be at that level given you're able to dedicate more full time hours in that role.