Ceiling fan + dimmable lights + old wiring?
Note: Depending on location and other factors (e.g., your lease) you may not be allowed to make any electrical system changes in a rented apartment. So any changes recommended by me or anyone else should be considered as "maybe I can do them myself or maybe I have to pay an electrician to do it".

Traditionally, a ceiling fan + light could be controlled one of two ways:


Two switches, with two switched hot wires. One switched hot would control the fan, one would control the lights and they would share a neutral. Putting a dimmer switch on the lighting only switch would work just fine, because it would control only the lights and not the fan. Based on your description (including "disconnected the fan motor"), I don't think you have this configuration.
One switch, with pull chains for the light and fan. This sounds like what you have. You can't use a dimmer in this configuration because whenever you dim lights you would cause big problems for the fan motor, as controlling speed of a fan requires a different kind of control than a normal dimmer.


There are two solutions:


Smart Switch - Get a smart switch that has a switch module with two switches, one including a dimmer, and with a remote box that connects to the fan/light. A smart switch can control multiple devices all over a single wire. If you want the option to connect "anything", that is the way to do it. But fortunately, new fans anticipate these requirements, so...
New Fan/Light with Remote - For example, picked at random Hunter Channing 54 in. LED Indoor Easy Install Brushed Nickel Ceiling Fan with HunterExpress feature set and checking the installation guide:


Remote controls fan speed and light dimming
Includes single-switch instructions on page 6 of the installation guide



Bottom line: Get a fan/light that includes a remote and single-switch instructions and you're all set.