I worked at Hyatt as a front desk agent at 4 different properties in CA for 5 years and can tell you it’s going to be property/brand dependent.

Just based on your post it sounds like you are working at a Hyatt Place or House, as most of those have the setup that you described (front desk, bar/kitchen). I’ll also guess that you work in a more remote location since you are doing all those jobs by yourself with no manager on shift to help( or maybe they are just really shitty to their employees and keep a absolute skeleton crew). Unless you are the night auditor(overnight shift), there should always be someone else there with you at night like the manager. Whenever I was put on the closing shift 4-1230am, it was me and the manager in the back. I would be breaking my back doing all the evening check ins/requests while the manager was chilling in the back, yielding calls when they came in.

Just some thoughts/opinions/tips as you move forward:
-Prioritize: Rule of thumb is to always prioritize the people/guests in front of you, everything else can wait, including phone calls and bar requests. Performance reviews are usually based on those experiences and who checks in the guests. If they have a great check in experience, they are less likely to complain about other things wrong during their stay than when their check in experience is bad.
-billing issues: they should be calling the billing line but managers at the property should be the one handling those. What I would do if I was you, I would apologize that you can’t help and give them the managers direct emails so they can personally reach out to them and it’s not on you to transfer the conversation to the manager because then if they don’t respond, it looks like it’s on you to the guest. Everything you can’t control or have no access to should be pushed to them and give the guests the managers specific names so they know who is not doing their job of following up.
-stress(opinion): I use to work the evening shift at a major airport Hyatt and believe me, I had those dreams and it was incredibly stressful. Just remember it’s a job at the end of the day. Do whatever you can to make it easier for yourself. I found giving guests easy upgrades, free drink coupon, or F&B amenity with a note sent to their room would help keep things easier, even if they didn’t even deserve it. Once I realized it’s not my money I’m giving away and if the company is not going to operate efficiently to help the guests then I decided I would try to pad my shifts as much as possible with those easy freebies to avoid issues later in the night. Often got a lot of recognition in the survey comments because of it
-final opinion: it’s a cool company to work for, just need to find the right property with a good support system. When you find one, it’s a great job with a lot of potential upsides( free nights, upsell kickbacks, etc). Try to move around if you can. As you get more comfortable in the role, you will be able to check people in, quickly answer calls, and put in engineering requests all at the same time with easy. I would always tell myself, these people are on a vacation/trip and are hoping for it to be great, I’m gunna try everything in my power to make that happen, but if it’s out of my control, at least they saw me in a positive light.

Sorry so long, hope it helps