Hi. : )
Union Station to LAX Hilton is a long trip - just so you know in advance! It'll be approximately an hour and 20 minutes if you are able to catch each bus (and if they're on time).
I tried looking on the Metro website and it looks like the Metro Rapid Line 745 [Harbor Fwy Station] heading south will take you to S Broadway / Century. You have to cross the street twice (the stop on the opposite side of the Shell station (there should be a parking lot). You want to be on W Century Blvd, not on S Broadway. Then, take the Metro Local Line 111 [LAX City Bus Ctr] heading west. You should see the LAX Hilton (25 stops).
I hope this helps! Just in case, you should ask someone at Union Station (and/or look at the Metro maps).
Best of luck~!