Tbilisi international airport is about 20-25 mins from the old town by taxi, very straightforward. Fixed price taxis from arrivals cost around 30-40 GEL which is not expensive, and Bolt works fine here too if you prefer to book through an app. The drive is quick and easy and you arrive directly in front of the hotel without any complicated logistics.
From the airport there is also an express bus that runs regularly and costs almost nothing, drops you near Freedom Square which is a short walk or quick taxi from the hotel. Takes a bit longer than a direct cab but perfectly fine if youre traveling light and not in a rush.
Getting around tbilisi itself on foot from this hotel is genuinely excellent, the old town is compact and most of what you want to see in the first few days is walkable. Metro exists and is cheap and covers the main corridors across the city, Bolt fills in the gaps for anything further. You really dont need a car in tbilisi and honestly having one in the old town would be more problem than solution.
Day trips out of the city are worth planning - Kazbegi in the north is one of the most spectacular mountain drives in the entire caucasus region, Kakheti wine region to the east is an easy half day, Mtskheta ancient capital is 20 mins away. Marshrutkas and shared taxis run to all of these from the main station, hotel staff can point you in the right direction.
Parking in the old town is genuinely difficult, narrow lanes and almost no dedicated spaces means if you arrived with a rental car you will spend real time and stress solving the parking problem every single day. Leave the car somewhere sensible outside the historic zone and use your feet, it will make your whole stay better!