What program is Neo using?

Based on the first words we can read during this scene, Neo is searching online papers for Morpheus.

The first time we get a good look at the computer screen, we can read 'Searching...' as part of the used software, and the large header 'Global Search', before it scrolls to 'Morpheus eludes Police at Heathrow Airport'. The last entry we see has the caption '******** (Morpheus?) International Manhunt Underway'.

The picture of Morpheus is actually superimposed (as can be easily seen in the video): it can either be a saved result, or a search parameter.

The algorithm is searching globally: the first characters, barely visible, are Chinese, the second search result is in English, the third is from An-Nahar, a "leading Arabic-language daily newspaper published in Lebanon", and later on it's browsing 'The Courier Press'.

As for whether it is an existing program or not, sources are hard to be found, leading me to believe it is not an existing program. This would make sense, since Neo is an accomplished hacker, "guilty of virtually every computer crime we have a law for" (as agent Smith phrases it), who might want to skip the user-friendly GUI's of established operating systems.
A similar question is on Fantasy SE, here, and the bottomline is that if this software were existing, it was probably well-documented.

Nevertheless, the monospace font can be identified as (very similar to) OCR Extended:

^{Sources:
https://en.wikipedia.org/wiki/An-Nahar
https://matrix.fandom.com/wiki/Room_101}

The matrix script simply describes it as a search engine.

The DARKNESS CRACKLES with phosphorescent energy, the word "searching" blazing in around us as we EMERGE FROM a computer screen.

The screen flickers with windowing data as a search engine runs with a steady relentless rhythm.

I would assume it was not a real bit of software but something created for the film to appear like it is searching.

As for what Neo is searching for, the script doesn't say but we could guess, based on the events that follow, it is performing a task for a client or crawling the internet for either Morpheus or The Matrix. My reasoning for the latter is what Neo and Trinity talk about when they meet.

TRINITY I know because I was once looking for the same thing, but when he found me he told me I wasn't really looking for him. I was looking for an answer.

There is a hypnotic quality to her voice and Neo feels the words, like a drug, seeping into him.

TRINITY It's the question that drives us, the question that brought you here. You know the question just as I did.

NEO What is the Matrix?

It can only be assumed he's doing port scans to find computers with open ports so he can hack into them. Despite that fact that the film was released in 1999, due to the sets it's difficult to determine if the film is actually set in 1999. He appears to be using Nmap to port scan, and at one point Trinity is trying to get an exit and she is definitely using Nmap to communicate with Tank.

It should be noted that any scenes where you see Matrix code (i.e. the green characters falling down the screen), that is definitely not legitimate code. That's all computer generated and a custom font meant to resemble Kanji.