The steps you listed will work, but there's a longer way that gives you more options:
git checkout dmgr2 # gets you "on branch dmgr2"
git fetch origin # gets you up to date with origin
git merge origin/master
The fetch command can be done at any point before the merge, i.e., you can swap the order of the fetch and the checkout, because fetch just goes over to the named remote (origin) and says to it: "gimme everything you have that I don't", i.e., all commits on all branches. They get copied to your repository, but named origin/branch for any branch named branch on the remote.
At this point you can use any viewer (git log, gitk, etc) to see "what they have" that you don't, and vice versa. Sometimes this is only useful for Warm Fuzzy Feelings ("ah, yes, that is in fact what I want") and sometimes it is useful for changing strategies entirely ("whoa, I don't want THAT stuff yet").
Finally, the merge command takes the given commit, which you can name as origin/master, and does whatever it takes to bring in that commit and its ancestors, to whatever branch you are on when you run the merge. You can insert --no-ff or --ff-only to prevent a fast-forward, or merge only if the result is a fast-forward, if you like.
When you use the sequence:
git checkout dmgr2
git pull origin master
the pull command instructs git to run git fetch, and then the moral equivalent of git merge origin/master. So this is almost the same as doing the two steps by hand, but there are some subtle differences that probably are not too concerning to you. (In particular the fetch step run by pull brings over only origin/master, and it does not update the ref in your repo:1 any new commits winds up referred-to only by the special FETCH_HEAD reference.)
If you use the more-explicit git fetch origin (then optionally look around) and then git merge origin/master sequence, you can also bring your own local master up to date with the remote, with only one fetch run across the network:
git fetch origin
git checkout master
git merge --ff-only origin/master
git checkout dmgr2
git merge --no-ff origin/master
for instance.
1This second part has been changed—I say "fixed"—in git 1.8.4, which now updates "remote branch" references opportunistically. (It was, as the release notes say, a deliberate design decision to skip the update, but it turns out that more people prefer that git update it. If you want the old remote-branch SHA-1, it defaults to being saved in, and thus recoverable from, the reflog. This also enables a new git 1.9/2.0 feature for finding upstream rebases.)