There could be a few things wrong, but I suspect that your new crank is ISO taper and your old spindle is JIS taper (or vice versa). Basically, the spindle shoves too far into the crank's square hole. So, getting yourself the right spindle will fix it. Make sure to get the same length with the correct taper- there are a bunch of different lengths and offset of spindles available.
No need to remove the fixed cup- just install the bearings from the other side using grease to hold them in place, then insert the spindle, then put the plastic accordian thing in (protects the bearings from crud and water coming down the seat tube), then put back the adjustable cup.