I am sure that all PCB designers have to do this all the time.
In fact, if you are designing a digital circuit with clock frequencies below about 50 MHz, you almost never will have to do signal integrity analysis to get a working design. And if you know what you are doing it is possible to design up to 1 or 2 GHz by using "best practices" rather than complex simulations.
I worked in an organization doing 1, 2, and 3 Gb/s designs and never saw a signal integrity tool in use until 2005 or so. (Although full-blown 3-d EM simulation was very occasionally used for very sticky problems)
However as the number of high-speed nets in your design increases, it's not always possible to stick to best practices everywhere, and then a simulation is valuable to indicate how much you can get away with.
Which software can be used to do this?
In contrast to what another answer said, SPICE and its derivatives are not well suited to this type of simulation. SPICE is designed for lumped-element analysis at the transistor level. In the situation you described you need to simulate a distributed element (a transmission line) and you're unlikely to have a transistor-level model of your source or load. Some SPICE-derived tools might have a signal-integrity tool bolted on, but it isn't typically what they're good at.
Signal itegrity tools generally use higher level models to reduce simulation time when simulating dozens or hundreds of distributed elements. And they can take inputs from IBIS models of the source and load ICs. These are standardized high-level macromodels that don't reveal details of the IC internals that the vendor might not want to share with all its customers.
Although I use Altium regularly I haven't used its signal integrity modelling tools. But from the description, they do seem to be IBIS-based rather than SPICE-like, so would probably work in many cases.
Another well-known tool is HyperLynx from Mentor Graphics. Cadence offers a tool set called Sigrity which I believe is similar but I have never seen or used.