I should think the train would be the best option.
If my memory serves me there is a station there.
The line passes through Salou, tarragona should be the next stop.
Have a look at RENFE site.

As your mums journey begins in Tarragona, you may get more replies on the Costa Dorada forum. Or try the Message Board on http://www.Salou.co.uk as a lot of the local Brits / Spanish view this site and have a lot of information to give and Salou is very close to Tarragona. Once you are in the site, click on the Union Jack flag and on the right hand side of the screen about three quarters of the way down the page is the Message Board Icon.

Just another thought, has your mum booked her flights yet? If not then Ryanair fly from Prestwick to Reus which is only about 10 miles away from Tarragona.

All the best.

Regards
Mark

Thanks for the info guys.

I've had a look at the alsa.es website but they timetable only goes up to end of Sep - don't know if the service terminates then.

Her flight's already booked.

GW