If you've worked through the majority of Apostol, then I highly recommend Hubbard and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms. (Get it straight from the publisher; much cheaper than elsewhere). It is written very well, with the reader in mind. You'll learn linear algebra and some other requisites to start with. From there, you learn a bit of very basic topology and you end up doing some differential geometry-esque stuff later. There is SO much in this book... you could study it for a couple of years (especially if you dig into the appendix). Moreover, despite its rigor, there are many applications (that are actually very interesting).
As well, there is some courseware from Harvard that uses this book (look for Math 23a,b and Math 25a,b). And there is a (partial) solutions manual floating around.
You might also like to read:
-Set Theory and Metric Spaces by Kaplansky
-anything from the New Math Library (from the MAA)
-Particularly Basic Inequalities, Geometric Transformations, or Mathematics of Choice.
-the linear algebra in Apostol (be sure to get Vol 2 also!)
-the whole second volume of Apostol!
-Discrete Mathematics by Biggs (get the first edition!)
-Finite Dimensional Vector Spaces by Halmos,
-Principles of Mathematical Analysis by Rudin with this and this and these awesome lectures.
-Algebra by Artin is amazing, but hard! Enjoy these lectures. Vinberg's algebra text is supposed to be amazing and in a similar flavor to Artin (but a bit more gentle).
And you might also like these (great) reading lists:
-PROMYS
-Chicago Undergraduate Mathematics Bibliography
The paragraph you refer to is about probably 50th and 60th, and I am not well aware of the book from that period. However, I would like to point out that starting from 1980 and till 1992 a series of math and physics books was published under the title "Библиотечка Кванта" (Kvant's library). Some of these books are translations of very insightful books, but most are written by big names such as Kolmogorov, Pontryagin, etc. You can find all the issues here. If someone is at school, likes physics and math, and reads Russian, this is a great read.
I would also recommend to check out the magazine Kvant. It has tons of wonderful problems with solutions.
About other books: Probably the series by I.M.Gelfand and co-authors is worth mentioning. These books were initiated and planned by Izrael Moiseevich, but written mostly by the co-authors. You can find some of them in English just going through the books by Gelfand.
Best Answer
Well, as an applied math student, I also love Russian books so much. I found most Russian mathematicians are also interested in writing books, so it may be convenient to search by the authors.
Here are some that I know: Geometry: S.P.Novikov (as you mentioned), Fomenko (he has many books, including a book on "geometric intuition" and a nice textbook, with Mischenko).
Mechanics: V.I.Arnold (many books, including the wonderful GTM64), Sedov (expert on fluid mechanics, book on continuum mechanics and dimensional analysis), Zorich (has a famous book on mathematical analysis) wrote a book named "Mathematical Analysis of Problems in the Natural Sciences". Landau (10 vol. on physics)
Analysis: for example, Kolmogolov and Fomin's book on functional analysis