I agree, you can take a taxi, but it will be a hike to hotel. Check to see if there is a shuttle service from Ann Arbor to hotel. It may cost $100 round trip to the train. There is a bus service you may be able to use as well.
I know this doesn't answer your question, but do you HAVE to fly into Chicago? Neither of the two airports are close to the Amtrak station. If you take a cab, it'll easily be $30-$40. If you take public transportation you will either have to take a cab from the el/subway to the station or walk almost a mile with your bags etc. Once you get on the train, it'll be 4-5 hours (several family members take this train, it's often delayed) from Chicago to Ann Arbor.
If you were able to fly into Detroit it'd be significantly easier and much closer.
In any case, good luck!
I second the earlier post to fly into Detroit. Spirit airlines runs some cheap flights on this route sometimes as low as $9, or even $40. This would save a ton of time and actually be cheaper than Amtrak.
Michigan Flyer bus is the only reasonably affordable ground transportation between DTW and the Ann Arbor-Ypsilanti area. From its nearest stop (probably Kensington Court near Briarwood) you'd need to take a cab, Uber or another AATA bus to your hotel.
If another bus turns out to be your answer, it may actually be easier to ride The Flyer all the way to Blake Transit Center since all area buses, somehow or another, originate from there.
You can see recent arrival times and delays, if you use this site:
Ann Arbor station code is ARB and I checked for a Saturday morning train, number 350. In the last week it was half hour late once and early three times.
Explanation for why this route in particular is often delayed in this article from a few years ago:http://www.nytimes.com/2012/05/08/us/chicago-train-congestion-slows-whole-country.html?_r=0