I am employed as a computer systems analyst. I am a landed immigrant of Canada but I never bothered to apply for Canadian citizenship. Am I eligible to apply for TN status?

Even assuming that you meet all of the other eligibility requirements, the TN category is a creation of NAFTA so it is only available to Canadian citizens. Landed immigrants of Canada are not eligible. However, you may be eligible for H-1B status. Check the database for H-1B requirements. You should look at the L and E categories as well, since they do not directly arise from NAFTA.

My company plans to send me to the United States under L-1 status. I am a Canadian landed immigrant. Can I apply at a port of entry?

No. The right to apply for L-1 at a port of entry arises from NAFTA. You may still apply for L-1 although your employer will have to file a petition for a nonimmigrant worker in the United States. You also require visa from a U.S. consulate unless you are a landed immigrant having a common nationality with Canada (i.e. United Kingdom). If you do have a common nationality with citizens of Canada, subject to certain exceptions, you can go directly to the port of entry once your petition has been approved.