Is it possible to use the Dubai Transit Hotel when not Transiting?

I am not sure how Dubai Airport is laid out, but normally you need a boarding pass to go from the arrivals area where your plane unloads to the departure area where the transit hotels are located. This transfer from arrival to departure for transit passengers is usually one way only.

While you maybe able to sneak through as JonathanReez suggested in the comments, how do you explain the need to return to arrivals to clear immigration and enter the country? Starting your visit in Dubai by lying to officials is never a good idea.

What about checked bags, they will be taken away as unclaimed if you sleep it off. Then you get to deal with tracking them down.

Plus transit hotels tend not to be every economical, charging dearly. How much would your hotel charge for another night or early check in?

As is common in many International airports around the world, Dubai has separate arrivals and departure areas.

The airport hotels are located in the departure area of the airport. As an incoming passenger, to get to the departure area you would need to pass through security, which requires you to show a boarding pass for an outgoing flight. Without a boarding pass, you will not be allowed through security, and will not be able to access the hotel.

So no, it is not possible to spend a few hours at the hotel on arrival unless you are in transit.

Dubai Airport is very much a 24x7 operation, and a result hotels in Dubai are used to guests arriving/departing early and/or late - I would suggest contacting your hotel and seeing if they can arrange an early check-in. Given the time you're arriving it's likely you'll need to pay, but it's also very likely that the price will be less than the airport hotel charges for a few hours.

As an aside, the rooms in the transit hotel (or at least, the room I had when I stayed there last year) was far from "small" - in fact I was impressed with the size of the room given that it was within the airport!