With only four hours in Chicago, I would be very leery about going anywhere too far.
That said, you can take a 151 or 156 bus from Union Station directly there (though, note that the 156 operates weekdays only). The 151 boards southbound on Canal, which is right in front of the "Great Hall." You can either exit from the Great Hall onto Canal, or you can take the escalators up from the Amtrak ticket counter area (not the one that leads to the food court, but the one that leads away from the Hertz rental car desk, directly across from the ticket counter) up to Canal, and cross the street. You can also take a 156, which boards northbound on Canal at Jackson (which is the south end of the station). Follow the same instructions as for the 151, but head south a half block (up the escalators from the ticket counter, out the door, turn left, go to the corner and cross the street).
Fare is $2 cash (no transfers issued), or you can buy a CTA transit card from the machines in the METRA (not Amtrak) ticket counter area, which is halfway between the Great Hall and the Amtrak ticket coutners. A single ride with a transit card on a bus is $1.75, and a transfer (good for two additional rides) is 25 cents (if used within 2 hours of initial boarding). Or you could buy a $5 one-day pass, if you don't want to worry about transfers, from a machine next to the transit card machine.
The 151 and 156 both take the same street through Lincoln Park (though I would recommend the 156, if traveling on a weekday, because the route is a little bit shorter, and it often gets better equipment). Look for the Armitage stop, listen for the automated announcements, or ask the driver to let you know when to get off.
The ride there takes about 35-40 minutes (maybe slightly longer on the 151, since it winds its way through State Street and Michigan Avenue), so, as I said earlier, you may not have that much time. Since CTA's service can get unreliable, depending on the time of day you travel (if it's midday, it *shouldn't* be too bad, but there are no guarantees in life), I would recommend leaving the zoo 1.5 hours before your scheduled departure (which, if you're taking the Empire Builder, is 2:15, so I'd say no later than 12:45). The 151 and 156 to Union Station board across the street from where you get off northbound. Same fare rules apply.
The 151 ends directly in front of Union Station. The 156 goes past Union Station on Adams Street (get off when you cross the Chicago River the second time; the first crossing will be on LaSalle Street, the second will be after you've traveled through downtown and turned onto Adams).
If you're coming in on another Amtrak train, keep in mind that your train might arrive late, which will cut into the amount of time you have available.
If you have any other questions, feel free to ask.