How to replace a section of counter-top?

Unless the island is exactly 2' wide you will not find stock laminate that will replicate it. I presume you mean laminate and not "linoleum". You can cover damaged laminate with another layer but it would require sanding the old surface, cutting a new sheet to the approximate size (you would need to measure and then special order it -- usually takes a few days for places like Home Depot to get it from the vendor.) Go to the HD or Lowes kitchen design department and get some sample chips that look the most like the counters in your place and take them home and find the closest match before ordering. You have to use a special stinky adhesive to get the new laminate to adhere, then you would need to use a router with a special bit to trim the edges flush with the edge of the countertop overhang.

It would be easier and probably cheaper to contact someone who does laminate counters and have them do this for you.

I'd go to Home Depot and get ceramic tile that matches the laminate. Not an exact match but coordinates well with the color of the other laminate. I'd remove the edge trim or just get a piece of 1/2 inch or 3/4 in plywood to replace the top, set and grout the tile, then add wood edging to it.

It would be an improvement over the laminate top to the island and the landlord should not complain. If Home Depot has a quartz stone scrap piece of countertop that would fit the island, you could use that as the top with no plywood underlay and no worries about setting down hot pans on it in the future.

Do not tile a counter top, for the last couple of decades more and more are doing that, the grout is impossible to keep clean and harbors bacteria out the ying yang. If you are at all handy, you can apply Laminated right over the existing crap, that is the cheapest way to go. If it is all one solid piece, the box stores may be able to give you an exact replacement. that you can just swap out. There are also specialty counter places that can come out and do it all for you.