The first thing to do is to check your configuration with:
rstudio-server verify-installation
a number of times when updating R or RStudio I have run into the same error as you have and get the following error message.
-bash-4.1$ sudo rstudio-server verify-installation
29 Sep 2015 18:24:11 [rserver] ERROR system error 98 (Address already in use); OCCURRED AT: rstudio::core::Error rstudio::core::http::initTcpIpAcceptor(rstudio::core::http::SocketAcceptorService<boost::asio::ip::tcp>&, const std::string&, const std::string&) /root/rstudio/src/cpp/core/include/core/http/TcpIpSocketUtils.hpp:103; LOGGED FROM: int main(int, char* const*) /root/rstudio/src/cpp/server/ServerMain.cpp:436
rstudio-server start/running, process 48632
Although I have never been able to figure out the cause, I can suggest the following workaround:
1. change the port /etc/rstudio/rserver.conf for example from 8787 to 8788
2. open the new ports in your firewall settings. (allow access to the new port in /etc/sysconfig/iptables)
3. update your firewall: sudo /sbin/service iptables restart
4. restart Rstudio server: sudo rstudio-server restart
This has worked for me each of the ~4-5 times this has happened. Although I am not 100% sure this can help with your use case, it may. As an alternative, if you can use containers in your AWS setup, you may be interested in a great off-the shelf docker image with the latest R/Rstudio.