How to quickly open an application in Lubuntu?

You have 3 options, Synapse, Kupfer or GNOME Do.

Synapse is a graphical launcher enabling you to start applications and also to quickly find and access relevant documents and files (by making use of the Zeitgeist engine).

sudo apt-get install synapse

Kupfer is a summoner/launcher in the style of Quicksilver or GNOME Do. It can search and browse your files, launch desired applications and object you need in a quicker way.

sudo apt-get install kupfer

Gnome Do Allows you to quickly search for many items present in your GNOME desktop environment (applications, Firefox bookmarks, files, artists and albums in Rhythmbox, Pidgin buddies, etc.) and perform commonly used actions on those items (Run, Open, Email, Chat, Play, etc.).

Although GNOME Do primarily targets the GNOME desktop, it also works under other desktop environments.

sudo apt-get install gnome-do

You may find synapse more eyecandy, kupfer is the one that needs less dependencies if that worries you. Gnome Do has also a plugin to be used as dock (not enabled by default) but needs hardware acceleration for it.

All three use or adapt to your current gtk theme so they all look fine.

Press Alt+F2, It shows the run dialog which is a good way to launch applications quickly.

You can change this keybard shortcut by editing ~/.config/openbox/lubuntu-rc.xml.