You can take the Metro but be advised that you need to switch train 2 times so it might be easier to just drive.

Buy a "Day Pass" on your TAP card - this is very important! LA's Metro does not have free transfers and unlimited rides like London's tube (or any major Metro system that I know of for that matter). Each time you switch lines, you have to pay another $1.50. So if you buy single ride fares, you will end up paying $4.50 one way or $9 round trip. If you buy a day pass, you can ride for $5 all day. After visiting Olvera St, you can take the Metro to Hollywood or Pasadena for no extra costs.

Directions:
From Hilton Garden Inn (which is located AT the Mariposa train station), take the Green line towards Norwalk.
Transfer to Blue line at Willowbrook station (you need to exit the Green line station and enter the Blue line station on a different level) - You need to TAP your card at the pylon when you enter the Blue line station even if you have Day Pass
Take Blue line towards 7th Street Metro Center.
Transfer to Red or Purple line at 7th Street Metro Center (go down stairs, no need to exit the station) - You need to TAP your card at the pylon when you enter the Red/Purple line level
Take Red or Purple line towards Union Station.
Exit Union Station front entrance. Olvera Street is across the street.

If you are driving, Union Station happens to be the nearest public parking... Although there are privately operated parking lots nearby as well.