Dugundji's Topology is a really fine book, much better than Munkres in my opinion (and Hatcher's opinion as well if you look here
As many other people, I find Munkres pedestrian and uninspiring.
Anyway, Dugundji is not modern, however most of the core of general topology is quite an established subject hence apart from minor changes of notation you should be fine. It is a comprehensive book and self-contained book. I find his writing very clear and his proofs not too terse or too clever.
While it starts from the basic, it avoids the doughnut-mug motivational blurb and starts with topological spaces in their earnest (rather than e.g. introducing the concepts first for metric spaces).
I am not familiar with the whole book, but from what I have seen so far some possible issues are
- the volume is out of print so you'd have to find a library/second hand copy
- exercises have no solutions
- he does not discuss separation axioms weaker than Hausdorff
- he does not discuss compact spaces which are not Hausdorff