What would you say is the oldest species still alive today? As in, something that's changed so little that you could show a reconstruction of a however-old fossil to someone and they'd go "oh, yeah, that's a ____". I know sharks are ridiculously old, but I'm guessing some kind of invertebrate probably has them beat.
There are plenty of groups of organisms whose modern forms still look very similar to their relatives from many tens or hundreds of millions of years ago – so-called “living fossils”. Examples would include things like horseshoe crabs, nautiluses, silverfish, scorpions, dragonflies, jellyfish, hagfish, sharks, sturgeons, coelacanths, tuataras, crocodylians, turtles, ferns, horsetails, redwoods, and mosses.
If something is already physically well-adapted to a stable ecological niche, and experiences
little environmental pressure to change, then they tend to stick with
the evolutionary equivalent of “if it ain’t broke, don’t fix it”.
As for a specific species, it’s actually really hard to tell for certain, especially
when it comes to fossils. We can label a fossil as being the “same”
species as a modern one based on physical appearance, but if we had a
living copy we might classify it very differently based on other factors
like genetics. (Plus there’s the problem that the entire concept of a biological “species” is really just a messy human construct with no clear consensus of definition.)
That said, under current species classification the tadpole shrimp Triops cancriformis doesn’t seem to have significantly changed for the last 200 million years, and some conifer trees like Araucaria araucana date back to similar ages.
But if we also include microbes here, then the clear winners are deep-sea sulfur-cycling bacteria. Modern ones are indistinguishable from fossils over 2 billion years old.