You can install curl as non-root using a prefix where you have write permissions.
Download curl sources, untar and cd into the extracted directory. Then
./configure --prefix=$HOME/usr
make
make install
and add this to you ~/.profile:
PATH="$HOME/usr/bin:$PATH"
export PATH
LD_LIBRARY_PATH="$HOME/usr/lib:$LD_LIBRARY_PATH"
export LD_LIBRARY_PATH
PKG_CONFIG_PATH="$HOME/usr/lib/pkgconfig:$PKG_CONFIG_PATH"
export PKG_CONFIG_PATH
MANPATH=$HOME/usr/share/man:$MANPATH
export MANPATH
Note, after you've setup such $HOME/usr in your ~/.profile once you can easily install most other packages to that prefix too.