You can see the town center of Innsbruck. You could go along the Inn river. 

Innsbruck isn’t a big city. And for free there is not so much to do. It is a very tourist city. 

If you have a specific question ask here below. 

When are you traveling there? It is a huge difference whether you are there in winter or summer…

city center you have seen within 1-2 hours. 

I’m travelling there in April

April is usually still not suitable for hiking up in the mountains as there might be still a lot of snow or at least snow fields, which can make hiking dangerous. Also mountain huts aren’t open during that time… so if you don’t wanna spend a lot of money, April isn’t the best time to visit Innsbruck. 
You might take a train to seefeld and hike there a little bit. 
in Innsbruck you can hike/walk up to the upper terminus of the Hungerbergbahn, from where you can get some free city views. Likely you can even hike up to Seegrube by April, from there even better views.

beside Schloss Ambras and if you are into ski jumping the Bergisel site might be interesting for you, but they are not for free.

For decent value food that will fill your stomach I recommend you Swing Kitchen (vegan burgers) right in the center of the city. It isn’t as cheap anymore as it was when they opened the first location in Vienna a few years ago but still good value IMO. 
another good option is Mundvoll in the center. They offer delicious bowls.
if you like Indian food, then Jaipur is a good choice. But price wise more expensive than swing kitchen and you should reserve if you wanna eat there 

Innsbruck is a great base if you have a rail pass as there are scenic rail routes in all directions. I spent seven days there with my wife in an air bnb earlier this year. As the posts above say there is not much to see or do that cannot be done in a day. There is an extensive and cheap public transport network that will get you up into the hills but as MartinM says snow and ice could make hiking dangerous. 

Cheap food for non-vegans: Ralway! The OeBB also has staff-restrts in all main cities and in I its just beside the main station-external people pay more as staff, but its still quite cheap compared to rest of town , though its german only.

What I often do is buy a dayticket for the local transit-tram&bus and ride around.

Main site is the house with the assumed golden roof.

BUT; you have a railpass, nah? Do the ride out+back via th Arlberg line =going west to the Swiss-thats the most impressive part daytime to see mountains.