What makes you want to go to North Riverside Mall?
Looking at the store directory, there isn't anything special there, it looks like a C or D grade mall - definitely nothing worth a special trip, imo.
http://www.northriversideparkmall.com/#!alphabetical/c18js
Are there particular stores that you really want to shop at?
State Street, which is easily accessible from Chicago Union Station, has a lot of stores in all price ranges - from Target to Macy's and everything in between.
Go to Google Maps and input '100 S State Street Chicago, IL' and then 'search nearby' for 'shopping' - you'll be surprised at the enormous amount of places to shop at.
In fact, use Google Maps Street View and walk the street and see for yourself.
A one day pass for the CTA is now $10 per person, a single fare is $2.25 (exact change) so unless you're going to take more than 4 rides, it's not worth it.
Catch the 151 bus (my favorite) outside of the old Union Station building (the one with the big columns) at the corner of Canal and Jackson.
Get off at Macy's on the corner of Washington and State.
To get back to Union Station, walk 3 blocks south on State Street from where you got off to Adams where Starbucks is.
Go around the corner on Adams and catch the 151 back to Union Station (the bus will say 151 Union Station on the front)
An FYI - the 151 bus starts at Union Station northbound and ends at Union Station southbound - the best bus to take if you don't want to get lost.
It also goes down Michigan Avenue and is worth a trip even if you don't shop there, just cross the street and take the bus back to Union Station when you're done strolling around.
