I’m not going to judge whether it’s worth the hassle to get from Barcelona to Santo Domingo de Silos, but…
Can you tell me which page you were looking at? I’ll see if it contains any useful information. I gather that you have already found information on the BCN-RGS flights?
There is indeed a bus, but it leaves from the bus station in Burgos itself. She would have to take a taxi or a bus from Burgos airport to the bus station. (Options are listed
here.)
The only regular Burgos – Santo Domingo line that I have found is operated by Arceredillo. (The line to Caleruega.)
Their web page shows that the bus leaves from Burgos on these times:
Mon to Thu: 5.30pm
Fri: 6.30pm
Sat: 2pm
No service on Sundays and public holidays.
For the return, the site only mentions departures from Santo Domingo at 8.10am. But it doesn’t clarify in what direction (Burgos or Caleruega) that is? (But based on the timing I assume it's Burgos...)
If budget is not an issue, she could take a taxi from the airport to Santo Domingo de Silos. According to
this website it should cost a little over 100€ one way.
Just to be clear, if never done any of these trips or used any of these services. I did spent some time in Burgos, but that was more than 10 years ago...