RS232 Cable
What Is RS232 Port? | Tom's Hardware Forum
Video via RS232 | PICAXE Forum
Can a standalone Arduino Uno be used for converting TTL to RS-232 or vice versa?
No it will not work with a VGA monitor.
If it is specifically marked as RS-232 (or with a string of one’s and zero’s), then it is a serial port for things like serial mice.
If it on the other hand, it is marked with Mon. or a rectangle, then it is a DE-9 connector for CGA or EGA monitors.
VGA monitors use a special D-sub 15-pin connector (DE-15F) and cannot be connected even with an adapter because the pinouts are completely different (because VGA is analog and works very differently from the older, digital CGA and EGA standards). You could try looking for a converter, but I for one have not seen or heard of any such thing, and would imagine it would be expensive anyway.
Your best bet is to check the local classifieds (especially online ones such as Kijiji or eBay Classified) to see if anyone nearby has a 9-pin monitor collecting dust in the closet. You could also put your own ad specifically asking for one.
Update:
Seeing your photo and the model of the motherboard, it is most definitely a serial port. The P4C800 does not have a 9-pin monitor connector (or even onboard VGA).
What you see is a generic bezel not specific to that motherboard. Cases used to come with several bezels of common variations, but because these days, motherboard backplanes come in a wide variety, they don’t and instead, motherboards come with a bezel specific to the board that can be snapped into a case. The one you see has both icons because it can be used for different motherboards and on some, instead of a serial port, there is a 15-pin VGA connector there (which has the same physical dimensions of a 9-pin serial port).
In summary, no, a serial port is not a video port and there is no way to use a monitor of any kind with it (even a 9-pin monitor, let alone 15-pin one). Since it does not have onboard video, you will need to install a video-card.

A 9 pin serial port is a communication port and CANNOT be used for a monitor. VGA is 15 pins and 15 pins only. Is what you have perhaps a DE-9 (also known as serial) connector? I do not believe there is an existing adapter for that type of conversion.
If you are interested in playing retro DOS games, then I would suggest investigating DOSBox. This should work with your current monitor.
The original IBM PC and clones did not use monitors that communicated through serial RS-232. Instead, RS-232 was primarily used for communication with modems and printers. The original IBM PC offered CGA graphics, which was output using a female DE-9 connector as outlined below.

The pinout is given below.

In a deleted comment, user Hearth pointed out that "EGA and Hercules graphics also used the same 9-pin connector." The EGA pinout can be found here and the Hercules pinout (which is the same as MDA) can be found here. User plugwash posted an answer which includes the Professional Graphics Controller (PGC) manufactured by IBM for PCs This is often referred to as the Professional Graphics Adapter (PGA), which also used the same 9-pin connector. The PGA pinout can be found here.
The monitor given in the question has a female 9 pin VGA connector. The pinout is given below.

This is not CGA, EGA, PGA or MDA (Hercules) compatible.
The latest VGA uses a DE-15 connector as outlined below.

The relationship between 9-pin and 15-pin VGA pinout is given below.

Connecting monitor in the question to a DisplayPort could be accomplished by the use of the following.
- 3ft DisplayPort to VGA Cable 28AWG Gold Plated, Black
This cable has a 15-pin VGA male connector. - DB9 Female to HD15 VGA Female Serial Adapter
This converts from 15-pin VGA to 9-pin VGA. - 3ft Serial DB9/DB9 Straight-thru RS232 Male to Male Cable
The specificatins state this cable is also good for "multisync video applications".
A cheaper alternative would be the following. However, this would leave a lot of hardware attached directly to the back of the monitor.
- DisplayPort to VGA Cable, 1920x1200
- DB9 Male to HD15 VGA Female Serial Adapter
Note that modern computers expect to digitally read the VGA monitors capabilities direct from the monitor over the connecting cables and adapters. This may not be possible with your monitor. So, beware than even if the cable and adapters can deliver the video signal, the computer may not emit a signal without first knowing the monitors capabilities.
References
- DOSBox
- Color Graphics Adapter
- Enhanced Graphics Adapter
- Hercules Graphics Card
- IBM Monochrome Display Adapter
- Computer Video connector pin-out
- VGA 9 Pin
- VGA 9 to 15 pin
- File:DE15 Connector Pinout.svg
- D-subminiature: Computer video output: DE-9 connectors
- IBM Color/Graphics Monitor Adapter
- IBM Color Display (5153)
Based on your description, the display seems to be a SyncMaster 3Ne (CVP4237P).
This link on Reddit is for the Samsung Syncmaster 4NE but it seems to have a similar 9 pin connector.
While you say you want a DisplayPort or HDMI connection, the are both digital connections and VGA is analog. So your best bet is a VGA cable like this one found on eBay:
- Universal RS232 DB9 9 Pin Male to VGA Video 15 Pin Male Adapter
