As a beginner should I study older chess games?


Best to study stuff designed for your level. Even if you were 2000 USCF, a book designed for someone level 2000 USCF would still just use relevent excerpts from higher level games to illustrate a point. There are some books that explain every move of a game, but most just highlight something from a GM level game that's relevant to what is being taught, and understandable by the target audience.
"... [annotated games are] infinitely more useful than bare game scores. However, annotated games vary widely in quality. Some are excellent study material. Others are poor. But the most numerous fall into a third category - good-but-wrong-for-you. ... You want games with annotations that answer the questions that baffle you the most. ..." - GM Andrew Soltis (2010)
Here are some reading possibilities that I often mention:
Simple Attacking Plans by Fred Wilson (2012)
https://web.archive.org/web/20140708090402/https://www.chesscafe.com/text/review874.pdf
https://dev.jeremysilman.com/shop/pc/Simple-Attacking-Plans-77p3731.htm
Logical Chess: Move by Move by Irving Chernev (1957)
https://web.archive.org/web/20140708104437/https://www.chesscafe.com/text/logichess.pdf
The Most Instructive Games of Chess Ever Played by Irving Chernev (1965)
https://chessbookreviews.wordpress.com/tag/most-instructive-games-of-chess-ever-played/

If you mean "chess games by the old masters" as opposed to "chess games by modern players", then yes. You will find games by Morphy, Tarrasch, Reti, Capablanca etc much more comprehensible and more instructive than games by modern players.
IM pfren and Spelenderwijs are correct. Studying the games of the old masters can teach you a lot, even when you're not an experienced player. But one word of warning: Don't look at game collections that were written for very strong players. Master love Gary Kasparov's series "My Great Predecessors," but those pages of detailed analysis will go right over your head. There are game collections aimed at beginning payers, like "Logical Chess: Move by Move," or "The Most Instructive Games of Chess Ever Played" bu Irving Chernev. Or "Winning Chess Brilliancies" by Yasser Seirawan. I'm sure that members here can suggest other books appropriate for your level.
As you go over these games, remember to frequently ask yourself "Why?" You will learn a lot by trying to answer that question. And when try, but you just can't answer that question. Post the position here and ask your question. Everyone will help
... "Winning Chess Brilliancies" by Yasser Seirawan ...
http://www.nystar.com/tamarkin/review1.htm
… Reti's "Masters of the Chessboard" ...
http://www.thechessmind.net/blog/2012/4/1/book-notice-richard-retis-masters-of-the-chessboard.html


Modern GM games are beyond the understanding of the everyday player.
Study the games of the old masters like Morphy, Andersen, and even before them. Philidor, Greco, etc.

When I coached a very successful high school chess team, I had my newbies focus on opening principles, middlegame tactics, and some endgame principles like The Opposition and the Pawn Square Rule. Most study shouild be about tactics.
A good, free book, written by 1920's World Chess Champion Jose Capablanca, is about 122 pages long and covers the basic things you need to know to be successful at competitve chess. It's legally free in a modern algebraic notation edition here:
https://www.sources.com/SSR/Docs/Capablanca-ChessFundamentals.pdf
"... 'Chess Fundamentals' ... does not deal so minutely as this book will with the things that beginners need to know. ..." - from Capablanca's A Primer of Chess
"... For let’s make no mistake, what ground Capablanca covers, he covers well. I enjoyed reading Capablanca’s presentation of even well-worn and standard positions. ...
Still, when compared with other instructional books for beginners and intermediate players, Capablanca’s Chess Fundamentals would not be my first choice. Other books cover the same or similar ground with a less confusing structure and more thoroughness. The following works come to mind as equal or in some ways superior: Lasker’s Common Sense in Chess; Znosko-Borovsky’s series of books; and Edward Lasker’s Chess Strategy. Later works that equal or surpass Chess Fundamentals would include Reuben Fine’s Chess the Easy Way and any number of Horowitz tomes.
Capablanca’s work has historical interest and value, of course, and for that reason alone belongs in any chess lover’s library. But there are better instructional books on the market. Certainly the works of Seirawan, Silman, Pandolfini, Polgar, Alburt, etc. are more accessible, speak a more modern idiom, and utilize advances in chess teaching and general pedagogy, etc. ..." - David Kaufman (2007)
https://web.archive.org/web/20131010102057/http://www.chesscafe.com/text/review564.pdf
It might be of interest to look at the table of contents of A COMPLETE CHESS COURSE by Antonio Gude: "... 1 The Basic Rules of Chess 7 ... 2 Your First Chess Games 23 … 3 Openings and Basic Principles 33 ... 4 Putting Your Pieces to Work 52 ... 5 Strategy and Tactics 76 ... 6 Endgame Play and Further Openings 106 … 7 Combinations and Tactical Themes 128 ... 8 Attacking Play 163 ... 9 Your First Opening Repertoire 194 …"
http://www.gambitbooks.com/pdfs/A_Complete_Chess_Course.pdf