When you hook it up directly like that, there will always be some lag due to some small processing delay from the Monitor because it's an LCD, even if the input is already digital. Kadano and I measured the lag in this kind of setup and it was rather low, less than a frame on average and when we playtested it, the setup felt quite playable but it's not as good as using Faster Melee (Dolphin) on a PC and playing in 120hz.
According to test results, the Acer GN246HL has particularly bad input lag for a 144hz monitor at an average of above 10ms. This might come as a surprise to you, let me explain:
Unfortunately, there are two components to the total input lag.
The signal processing delay and the pixel response time. The first is the time it takes for the LCD to process the signal and tell the pixels how to move. The latter is the time it takes for the pixels to move once they are told to move.
The thing you see advertised on the Monitor as "1ms" is only the pixel response time and there are multiple ways of measuring response time. The Acer GN246HL unfortunately has particularly bad signal processing delay and a much lower than advertised pixel response time in all tests, except one.
Below you can find graphs detailing the reponse time and processing delay as tested by the reputable german website prad.de. Just by looking at the graphs, I am sure you can see the difference between the two monitors. You can also check out the source - it is in german but you can use google translate or see if the review is also available in english as they do translate some of their articles.