How can I make apt-get upgrade the kernel, similarly to aptitude?

According to the apt-get man page:

upgrade is used to install the newest versions of all packages currently installed on the system.

dist-upgrade, in addition to performing the function of upgrade, also intelligently handles changing dependencies with new versions of packages.

When you use aptitude, it automatically use the dependencies of the packages, if you want likely the same results, use apt-get dist-upgrade.

If you want to know the differences between the two, check this post or this one.

open your terminal and type this

sudo apt-get update
sudo apt-get dist-upgrade